Enhanced parimutuel wagering

ABSTRACT

Methods and systems for engaging in enhanced parimutuel wagering and gaming. In one embodiment, different types of bets can be offered and processed in the same betting pool on an underlying event, such as a horse or dog race, a sporting event or a lottery, and the premiums and payouts of these different types of bets can be determined in the same betting pool, by configuring an equivalent combination of fundamental bets for each type of bet, and performing a demand-based valuation of each of the fundamental bets in the equivalent combination. In another embodiment, bettors can place bets in the betting pool with limit odds on the selected outcome of the underlying event. The bets with limit odds are not filled in whole or in part, unless the final odds on the selected outcome of the underlying event are equal to or greater than the limit odds.

RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.12/660,400, filed Feb. 24, 2010; which is a continuation of U.S. patentapplication Ser. No. 10/640,656, filed Aug. 13, 2003, which issued asU.S. Pat. No. 7,742,972; which is a continuation-in-part of U.S. patentapplication Ser. No. 10/365,033, filed Feb. 11, 2003; which is acontinuation-in-part of U.S. patent application Ser. No. 10/115,505,filed Apr. 2, 2002; which is a continuation-in-part of U.S. patentapplication Ser. No. 09/950,498, filed Sep. 10, 2001; which is acontinuation-in-part of U.S. patent application Ser. No. 09/809,025,filed Mar. 16, 2001, which issued as U.S. Pat. No. 7,225,153; which is acontinuation-in-part of U.S. patent application Ser. No. 09/744,816initially filed Jan. 31, 2001 and attributed a filing date of Apr. 3,2001, which issued as U.S. Pat. No. 7,389,262, and which is the UnitedStates national stage application under 35 U.S.C. §371 of PatentCooperation Treaty Application No. PCT/US00/19447, filed Jul. 18, 2000;which claims priority to U.S. patent application Ser. No. 09/448,822,filed Nov. 24, 1999, which issued as U.S. Pat. No. 6,321,212; whichclaims the benefit, under 35 U.S.C. §119(e), of U.S. Provisional PatentApplication Ser. No. 60/144,890, filed Jul. 21, 1999. Each of theapplications referred to in this paragraph is incorporated herein byreference in its entirety.

COPYRIGHT NOTICE

This document contains material that is subject to copyright protection.The applicant has no objection to the facsimile reproduction of thispatent document, as it appears in the U.S. Patent and Trademark Office(PTO) patent file or records or in any publication by the PTO orcounterpart foreign or international instrumentalities. The applicantotherwise reserves all copyright rights whatsoever.

FIELD OF THE INVENTION

This invention relates to systems and methods for enhanced parimutuelwagering and gaming. More specifically, this invention relates tomethods and systems for enabling different types of bets to be offeredin the same betting pool, and for determining the premiums and payoutsof these different types of bets in the same betting pool, byconfiguring an equivalent combination of fundamental bets for each typeof bet, and performing a demand-based valuation of each of thefundamental bets in the equivalent combination. This invention alsorelates to methods and systems for enabling and determining values forbets placed in the betting pool with limit odds on the selected outcomeof the underlying wagering event. The bets with limit odds are notfilled in whole or in part, unless the final odds on the selectedoutcome of the underlying event are equal to or greater than the limitodds.

BACKGROUND OF THE INVENTION

With the rapid increase in usage and popularity of the public Internet,the growth of electronic Internet-based trading of securities has beendramatic. In the first part of 1999, online trading via the Internet wasestimated to make up approximately 15% of all stock trades. This volumehas been growing at an annual rate of approximately 50%. High growthrates are projected to continue for the next few years, as increasingvolumes of Internet users use online trading accounts.

Online trading firms such as E-Trade Group, Charles Schwab, andAmeritrade have all experienced significant growth in revenues due toincreases in online trading activity. These companies currently offerInternet-based stock trading services, which provide greater convenienceand lower commission rates for many retail investors, compared totraditional securities brokerage services. Many expect online trading toexpand to financial products other than equities, such as bonds, foreignexchange, and financial instrument derivatives.

Financial products such as stocks, bonds, foreign exchange contracts,exchange traded futures and options, as well as contractual assets orliabilities such as reinsurance contracts or interest-rate swaps, allinvolve some measure of risk. The risks inherent in such products are afunction of many factors, including the uncertainty of events, such asthe Federal Reserve's determination to increase the discount rate, asudden increase in commodity prices, the change in value of anunderlying index such as the Dow Jones Industrial Average, or an overallincrease in investor risk aversion. In order to better analyze thenature of such risks, financial economists often treat the real-worldfinancial products as if they were combinations of simpler, hypotheticalfinancial products. These hypothetical financial products typically aredesigned to pay one unit of currency, say one dollar, to the trader orinvestor if a particular outcome among a set of possible outcomesoccurs. Possible outcomes may be said to fall within “states,” which aretypically constructed from a distribution of possible outcomes (e.g.,the magnitude of the change in the Federal Reserve discount rate) owingto some real-world event (e.g., a decision of the Federal Reserveregarding the discount rate). In such hypothetical financial products, aset of states is typically chosen so that the states are mutuallyexclusive and the set collectively covers or exhausts all possibleoutcomes for the event. This arrangement entails that, by design,exactly one state always occurs based on the event outcome.

These hypothetical financial products (also known as Arrow-Debreusecurities, state securities, or pure securities) are designed toisolate and break-down complex risks into distinct sources, namely, therisk that a distinct state will occur. Such hypothetical financialproducts are useful since the returns from more complicated securities,including real-world financial products, can be modeled as a linearcombination of the returns of the hypothetical financial products. See,e.g., R. Merton, Continuous-Time Finance (1990), pp. 441 ff. Thus, suchhypothetical financial products are frequently used today to provide thefundamental building blocks for analyzing more complex financialproducts.

In recent years, the growth in derivatives trading has also beenenormous. According to the Federal Reserve, the annualized growth ratein foreign exchange and interest rate derivatives turnover alone isrunning at about 20%. Corporations, financial institutions, farmers, andeven national governments and agencies are all active in the derivativesmarkets, typically to better manage asset and liability portfolios,hedge financial market risk, and minimize costs of capital funding.Money managers also frequently use derivatives to hedge and undertakeeconomic exposure where there are inherent risks, such as risks offluctuation in interest rates, foreign exchange rates, convertibilityinto other securities or outstanding purchase offers for cash orexchange offers for cash or securities.

Derivatives are traded on exchanges, such as the option and futurescontracts traded on the Chicago Board of Trade (“CBOT”), as well asoff-exchange or over-the-counter (“OTC”) between two or more derivativecounterparties. On the major exchanges that operate trading activity inderivatives, orders are typically either transmitted electronically orvia open outcry in pits to member brokers who then execute the orders.These member brokers then usually balance or hedge their own portfolioof derivatives to suit their own risk and return criteria. Hedging iscustomarily accomplished by trading in the derivatives' underlyingsecurities or contracts (e.g., a futures contract in the case of anoption on that future) or in similar derivatives (e.g., futures expiringin different calendar months). For OTC derivatives, brokers or dealerscustomarily seek to balance their active portfolios of derivatives inaccordance with the trader's risk management guidelines andprofitability criteria.

Broadly speaking then, there are two widely utilized means by whichderivatives are currently traded: (1) order-matching and (2) principalmarket making. Order matching is a model followed by exchanges such asthe CBOT or the Chicago Mercantile Exchange and some newer onlineexchanges. In order matching, the exchange coordinates the activities ofbuyers and sellers so that “bids” to buy (i.e., demand) can be pairedoff with “offers” to sell (i.e., supply). Orders may be matched bothelectronically and through the primary market making activities of theexchange members. Typically, the exchange itself takes no market riskand covers its own cost of operation by selling memberships to brokers.Member brokers may take principal positions, which are often hedgedacross their portfolios.

In principal market making, a bank or brokerage firm, for example,establishes a derivatives trading operation, capitalizes it, and makes amarket by maintaining a portfolio of derivatives and underlyingpositions. The market maker usually hedges the portfolio on a dynamicbasis by continually changing the composition of the portfolio as marketconditions change. In general, the market maker strives to cover itscost of operation by collecting a bid-offer spread and through the scaleeconomies obtained by simultaneously hedging a portfolio of positions.As the market maker takes significant market risk, its counterpartiesare exposed to the risk that it may go bankrupt. Additionally, while intheory the principal market making activity could be done over a widearea network, in practice derivatives trading is today usuallyaccomplished via the telephone. Often, trades are processed laboriously,with many manual steps required from the front office transaction to theback office processing and clearing.

In theory—that is, ignoring very real transaction costs (describedbelow)—derivatives trading is, in the language of game theory, a “zerosum” game. One counterparty's gain on a transaction should be exactlyoffset by the corresponding counterparty's loss, assuming there are notransaction costs. In fact, it is the zero sum nature of the derivativesmarket which first allowed the well-known Black-Scholes pricing model tobe formulated by noting that a derivative such as an option could bepaired with an exactly offsetting position in the underlying security soas to eliminate market risk over short periods of time. It is this “noarbitrage” feature that allows market participants using sophisticatedvaluation models to mitigate market risk by continually adjusting theirportfolios. Stock markets, by contrast, do not have this zero sumfeature, as the total stock or value of the market fluctuates due tofactors such as interest rates and expected corporate earnings, whichare “external” to the market in the sense that they cannot readily behedged.

The return to a trader of a traditional derivative product is, in mostcases, largely determined by the value of the underlying security,asset, liability or claim on which the derivative is based. For example,the value of a call option on a stock, which gives the holder the rightto buy the stock at some future date at a fixed strike price, variesdirectly with the price of the underlying stock. In the case ofnon-financial derivatives such as reinsurance contracts, the value ofthe reinsurance contract is affected by the loss experience on theunderlying portfolio of insured claims. The prices of traditionalderivative products are usually determined by supply and demand for thederivative based on the value of the underlying security (which isitself usually determined by supply and demand, or, as in the case ofinsurance, by events insured by the insurance or reinsurance contract).

At present, market-makers can offer derivatives products to theircustomers in markets where:

-   -   Sufficient natural supply and demand exist    -   Risks are measurable and manageable    -   Sufficient capital has been allocated        A failure to satisfy one or more of these conditions in certain        capital markets may inhibit new product development, resulting        in unsatisfied customer demand.

Currently, the costs of trading derivative securities (both on and offthe exchanges) and transferring insurance risk are considered to be highfor a number of reasons, including:

-   (1) Credit Risk: A counterparty to a derivatives (or insurance    contract) transaction typically assumes the risk that its    counterparty will go bankrupt during the life of the derivatives (or    insurance) contract. Margin requirements, credit monitoring, and    other contractual devices, which may be costly, are customarily    employed to manage derivatives and insurance counterparty credit    risk.-   (2) Regulatory Requirements: Regulatory bodies, such as the Federal    Reserve, Comptroller of the Currency, the Commodities Futures    Trading Commission, and international bodies that promulgate    regulations affecting global money center banks (e.g., Basle    Committee guidelines) generally require institutions dealing in    derivatives to meet capital requirements and maintain risk    management systems. These requirements are considered by many to    increase the cost of capital and barriers to entry for some entrants    into the derivatives trading business, and thus to increase the cost    of derivatives transactions for both dealers and end users. In the    United States, state insurance regulations also impose requirements    on the operations of insurers, especially in the property-casualty    lines where capital demands may be increased by the requirement that    insurers reserve for future losses without regard to interest rate    discount factors.-   (3) Liquidity: Derivatives traders typically hedge their exposures    throughout the life of the derivatives contract. Effective hedging    usually requires that an active or liquid market exist, throughout    the life of the derivative contract, for both the underlying    security and the derivative. Frequently, especially in periods of    financial market shocks and disequilibria, liquid markets do not    exist to support a well-functioning derivatives market.-   (4) Transaction Costs: Dynamic hedging of derivatives often requires    continual transactions in the market over the life of the derivative    in order to reduce, eliminate, and manage risk for a derivative or    portfolio of derivative securities. This usually means paying    bid-offers spreads for each hedging transaction, which can add    significantly to the price of the derivative security at inception    compared to its theoretical price in absence of the need to pay for    such spreads and similar transaction costs.-   (5) Settlement and Clearing Costs: The costs of executing,    electronically booking, clearing, and settling derivatives    transactions can be large, sometimes requiring analytical and    database software systems and personnel knowledgeable in such    transactions. While a goal of many in the securities processing    industry is to achieve “straight-through-processing” of derivatives    transactions, many derivatives counterparties continue to manage the    processing of these transactions using a combination of electronic    and manual steps which are not particularly integrated or automated    and therefore add to costs.-   (6) Event Risk: Most traders understand effective hedging of    derivatives transactions to require markets to be liquid and to    exhibit continuously fluctuating prices without sudden and dramatic    “gaps.” During periods of financial crises and disequilibria, it is    not uncommon to observe dramatic repricing of underlying securities    by 50% or more in a period of hours. The event risk of such crises    and disequilibria are therefore customarily factored into    derivatives prices by dealers, which increases the cost of    derivatives in excess of the theoretical prices indicated by    derivatives valuation models. These costs are usually spread across    all derivatives users.-   (7) Model Risk: Derivatives contracts can be quite difficult to    value, especially those involving interest rates or features which    allow a counterparty to make decisions throughout the life of the    derivative (e.g., American options allow a counterparty to realize    the value of the derivative at any time during its life).    Derivatives dealers will typically add a premium to derivatives    prices to insure against the possibility that the valuation models    may not adequately reflect market factors or other conditions    throughout the life of the contract. In addition, risk management    guidelines may require firms to maintain additional capital    supporting a derivatives dealing operation where model risk is    determined to be a significant factor. Model risk has also been a    large factor in well-known cases where complicated securities risk    management systems have provided incorrect or incomplete    information, such as the Joe Jett/Kidder Peabody losses of 1994.-   (8) Asymmetric Information: Derivatives dealers and market makers    customarily seek to protect themselves from counterparties with    superior information. Bid-offer spreads for derivatives therefore    usually reflect a built-in insurance premium for the dealer for    transactions with counterparties with superior information, which    can lead to unprofitable transactions. Traditional insurance markets    also incur costs due to asymmetric information. In property-casualty    lines, the direct writer of the insurance almost always has superior    information regarding the book of risks than does the assuming    reinsurer. Much like the market maker in capital markets, the    reinsurer typically prices its ° informational disadvantage into the    reinsurance premiums.-   (9) Incomplete Markets: Traditional capital and insurance markets    are often viewed as incomplete in the sense that the span of    contingent claims is limited, i.e., the markets may not provide    opportunities to hedge all of the risks for which hedging    opportunities are sought. As a consequence, participants typically    either bear risk inefficiently or use less than optimal means to    transfer or hedge against risk. For example, the demand by some    investors to hedge inflation risk has resulted in the issuance by    some governments of inflation-linked bonds which have coupons and    principal amounts linked to Consumer Price Index (CPI) levels. This    provides a degree of insurance against inflation risk. However,    holders of such bonds frequently make assumptions as to the future    relationship between real and nominal interest rates. An imperfect    correlation between the contingent claim (in this case,    inflation-linked bond) and the contingent event (inflation) gives    rise to what traders call “basis risk,” which is risk that, in    today's markets, cannot be perfectly insured or hedged.

Currently, transaction costs are also considerable in traditionalinsurance and reinsurance markets. In recent years, considerable efforthas been expended in attempting to securitize insurance risk such asproperty-casualty catastrophe risk. Traditional insurance andreinsurance markets in many respects resemble principal market-makersecurities markets and suffer from many of the same shortcomings andincur similar costs of operation. Typically, risk is physicallytransferred contractually, credit status of counterparties is monitored,and sophisticated risk management systems are deployed and maintained.Capitalization levels to support insurance portfolios of risky assetsand liabilities may be dramatically out of equilibrium at any given timedue to price stickiness, informational asymmetries and costs, andregulatory constraints. In short, the insurance and reinsurance marketstend to operate according to the same market mechanisms that haveprevailed for decades, despite large market shocks such as the Lloydscrisis in the late 1980's and early 1990's.

Accordingly, a driving force behind all the contributors to the costs ofderivatives and insurance contracts is the necessity or desirability ofrisk management through dynamic hedging or contingent claim replicationin continuous, liquid, and informationally fair markets. Hedging is usedby derivatives dealers to reduce their exposure to excessive market riskwhile making transaction fees to cover their cost of capital and ongoingoperations; and effective hedging requires liquidity.

Recent patents have addressed the problem of financial market liquidityin the context of an electronic order-matching systems (e.g., U.S. Pat.No. 5,845,266). The principal techniques disclosed to enhance liquidityare to increase participation and traded volume in the system and tosolicit trader preferences about combinations of price and quantity fora particular trade of a security. There are shortcomings to thesetechniques, however. First, these techniques implement order-matchingand limit order book algorithms, which can be and are effectivelyemployed in traditional “brick and mortar” exchanges. Their electronicimplementation, however, primarily serves to save on transportation andtelecommunication charges. No fundamental change is contemplated tomarket structure for which an electronic network may be essential.Second, the disclosed techniques appear to enhance liquidity at theexpense of placing large informational burdens on the traders (bysoliciting preferences, for example, over an entire price-quantitydemand curve) and by introducing uncertainty as to the exact price atwhich a trade has been transacted or is “filled.” Finally, theseelectronic order matching systems contemplate a traditional counterpartypairing, which means physical securities are frequently transferred,cleared, and settled after the counterparties are identified andmatched. In other words, techniques disclosed in the context ofelectronic order-matching systems are technical elaborations to thebasic problem of how to optimize the process of matching arrays of bidsand offers.

Patents relating to derivatives, such as U.S. Pat. No. 4,903,201,disclose an electronic adaptation of current open-outcry or ordermatching exchanges for the trading of futures is disclosed. Anotherrecent patent, U.S. Pat. No. 5,806,048, relates to the creation ofopen-end mutual fund derivative securities to provide enhanced liquidityand improved availability of information affecting pricing. This patent,however, does not contemplate an electronic derivatives exchange whichrequires the traditional hedging or replicating portfolio approach tosynthesizing the financial derivatives. Similarly, U.S. Pat. No.5,794,207 proposes an electronic means of matching buyers' bids andsellers' offers, without explaining the nature of the economic priceequilibria achieved through such a market process.

SUMMARY OF THE INVENTION

The present invention is directed to systems and methods of trading, andfinancial products, having a goal of reducing transaction costs formarket participants who hedge against or otherwise make investments incontingent claims relating to events of economic significance. Theclaims are contingent in that their payout or return depends on theoutcome of an observable event with more than one possible outcome. Anexample of such a contingent claim is a digital option, such as adigital call option, where the investor receives a payout if theunderlying asset, stock or index expires at or above a specified strikeprice and receives no payout if the underlying asset, stock or otherindex expires below the strike price. Digital options can also bereferred to as, for example, “binary options” and “all or nothingoptions.” The contingent claims relate to events of economicsignificance in that an investor or trader in a contingent claimtypically is not economically indifferent to the outcome of the event,even if the investor or trader has not invested in or traded acontingent claim relating to the event.

Intended users of preferred and other embodiments of the presentinvention are typically institutional investors, such as financialinstitutions including banks, investment banks, primary insurers andreinsurers, and corporate treasurers, hedge funds and pension funds.Users can also include any individual or entity with a need for riskallocation services. As used in this specification, the terms “user,”“trader” and “investor” are used interchangeably to mean anyinstitution, individual or entity that desires to trade or invest incontingent claims or other financial products described in thisspecification.

The contingent claims pertaining to an event have a trading period or anauction period in order to finalize a return for each defined state,each defined state corresponding to an outcome or set of outcomes forthe event, and another period for observing the event upon which thecontingent claim is based. When the contingent claim is a digitaloption, the price or investment amount for each digital option isfinalized at the end of the trading period, along with the return foreach defined state. The entirety of trades or orders placed and acceptedwith respect to a certain trading period are processed in a demand-basedmarket or auction. The organization or institution, individual or otherentity sponsoring, running, maintaining or operating the demand-basedmarket or auction, can be referred to, for example, as an “exchange,”“auction sponsor” and/or “market sponsor.”

In each market or auction, the returns to the contingent claims adjustduring the trading period of the market or auction with changes in thedistribution of amounts invested in each of the states. The investmentamounts for the contingent claims can either be provided up front ordetermined during the trading period with changes in the distribution ofdesired returns and selected outcomes for each claim. The returnspayable for each of the states are finalized after the conclusion ofeach relevant trading period. In a preferred embodiment, the totalamount invested, less a transaction fee to an exchange, or a market orauction sponsor, is equal to the total amount of the payouts. In otherwords, in theory, the returns on all of the contingent claimsestablished during a particular trading period and pertaining to aparticular event are essentially zero sum, as are the traditionalderivatives markets. In one embodiment, the investment amounts or pricesfor each contingent claim are finalized after the conclusion of eachrelevant trading period, along with the returns payable for each of thestates. Since the total amount invested, less a transaction fee to anexchange, or a market or auction sponsor, is equal to the total amountof payouts, an optimization solution using an iteration algorithmdescribed below can be used to determine the equilibrium investmentamounts or prices for each contingent claim along with establishing thereturns on all of the contingent claims, given the desired or requestedreturn for each claim, the selection of outcomes for each claim and thelimit (if any) on the investment amount for each claim.

The process by which returns and investment amounts for each contingentclaim are finalized in the present invention is demand-based, and doesnot in any substantial way depend on supply. By contrast, traditionalmarkets set prices through the interaction of supply and demand bycrossing bids to buy and offers to sell (“bid/offer”). The demand-basedcontingent claim mechanism of the present invention sets returns byfinancing returns to successful investments with losses fromunsuccessful investments. Thus, in a preferred embodiment, the returnsto successful investments (as well as the prices or investment amountsfor investments in digital options) are determined by the total andrelative amounts of all investments placed on each of the defined statesfor the specified observable event.

As used in this specification, the term “contingent claim” shall havethe meaning customarily ascribed to it in the securities, trading,insurance and economics communities. “Contingent claims” thus include,for example, stocks, bonds and other such securities, derivativesecurities, insurance contracts and reinsurance agreements, and anyother financial products, instruments, contracts, assets, or liabilitieswhose value depends upon or reflects economic risk due to the occurrenceof future, real-world events. These events may be financial-relatedevents, such as changes in interest rates, or non-financial-relatedevents such as changes in weather conditions, demand for electricity,and fluctuations in real estate prices. Contingent claims also includeall economic or financial interests, whether already traded or not yettraded, which have or reflect inherent risk or uncertainty due to theoccurrence of future real-world events. Examples of contingent claims ofeconomic or financial interest which are not yet traded on traditionalmarkets are financial products having values that vary with thefluctuations in corporate earnings or changes in real estate values andrentals. The term “contingent claim” as used in this specificationencompasses both hypothetical financial products of the Arrow-Debreuvariety, as well as any risky asset, contract or product which can beexpressed as a combination or portfolio of the hypothetical financialproducts.

For the purposes of this specification, an “investment” in or “trade” oran “order” of a contingent claim is the act of putting an amount (in theunits of value defined by the contingent claim) at risk, with afinancial return depending on the outcome of an event of economicsignificance underlying the group of contingent claims pertaining tothat event.

“Derivative security” (used interchangeably with “derivative”) also hasa meaning customarily ascribed to it in the securities, trading,insurance and economics communities. This includes a security orcontract whose value depends on such factors as the value of anunderlying security, index, asset or liability, or on a feature of suchan underlying security, such as interest rates or convertibility intosome other security. A derivative security is one example of acontingent claim as defined above. Financial futures on stock indicessuch as the S&P 500 or options to buy and sell such futures contractsare highly popular exchange-traded financial derivatives. Aninterest-rate swap, which is an example of an off-exchange derivative,is an agreement between two counterparties to exchange series ofcashflows based on underlying factors, such as the London InterbankOffered Rate (LIBOR) quoted daily in London for a large number offoreign currencies. Like the exchange-traded futures and options,off-exchange agreements can fluctuate in value with the underlyingfactors to which they are linked or derived. Derivatives may also betraded on commodities, insurance events, and other events, such as theweather.

In this specification, the function for computing and allocating returnsto contingent claims is termed the Demand Reallocation Function (DRF). ADRF is demand-based and involves reallocating returns to investments ineach state after the outcome of the observable event is known in orderto compensate successful investments from losses on unsuccessfulinvestments (after any transaction or exchange fee). Since an adjustablereturn based on variations in amounts invested is a key aspect of theinvention, contingent claims implemented using a DRF will be referred toas demand-based adjustable return (DBAR) contingent claims.

In accordance with embodiments of the present invention, an Order PriceFunction (OPF) is a function for computing the investment amounts orprices for contingent claims which are digital options. An OPF, whichincludes the DRF, is also demand-based and involves determining theprices for each digital option at the end of the trading period, butbefore the outcome of the observable event is known. The OPF determinesthe prices as a function of the outcomes selected in each digital option(corresponding to the states selected by a trader for the digital optionto be in-the-money), the requested payout for the digital option if theoption expires in-the money, and the limit placed on the price (if any)when the order for the option is placed in the market or auction.

“Demand-based market,” “demand-based auction” may include, for example,a market or auction which is run or executed according to the principlesset forth in the embodiments of the present invention. “Demand-basedtechnology” may include, for example, technology used to run or executeorders in a demand-based market or auction in accordance with theprinciples set forth in the embodiments of the present invention.“Contingent claims” or “DBAR contingent claims” may include, forexample, contingent claims that are processed in a demand-based marketor auction. “Contingent claims” or “DBAR contingent claims” may include,for example, digital options or DBAR digital options, discussed in thisspecification. With respect to digital options, demand-based markets mayinclude, for example, DBAR DOEs (DBAR Digital Option Exchanges), orexchanges in which orders for digital options or DBAR digital optionsare placed and processed. “Contingent claims” or “DBAR contingentclaims” may also include, for example, DBAR-enabled products orDBAR-enabled financial products, discussed in this specification.

Preferred features of a trading system for a group of DBAR contingentclaims (i.e., group of claims pertaining to the same event) include thefollowing: (1) an entire distribution of states is open for investment,not just a single price as in the traditional markets; (2) returns areadjustable and determined mathematically based on invested amounts ineach of the states available for investment, (3) invested amounts arepreferably non-decreasing (as explained below), providing a commitmentof offered liquidity to the market over the distribution of states, andin one embodiment of the present invention, adjustable and determinedmathematically based on requested returns per order, selection ofoutcomes for the option to expire in-the-money, and limit amounts (ifany), and (4) information is available in real-time across thedistribution of states, including, in particular, information on theamounts invested across the distribution of all states (commonly knownas a “limit order book”). Other consequences of preferred embodiments ofthe present invention include (1) elimination of order-matching orcrossing of the bid and offer sides of the market; (2) reduction of theneed for a market maker to conduct dynamic hedging and risk management;(3) more opportunities for hedging and insuring events of economicsignificance (i.e., greater market “completeness”); and (4) the abilityto offer investments in contingent claims whose profit and lossscenarios are comparable to these for digital options or otherderivatives in traditional markets, but can be implemented using theDBAR systems and methods of the present invention, for example withoutthe need for sellers of such options or derivatives as they function inconventional markets.

Other preferred embodiments of the present invention can accommodaterealization of profits and losses by traders at multiple points beforeall of the criteria for terminating a group of contingent claims areknown. This is accomplished by arranging a plurality of trading periods,each having its own set of finalized returns. Profit or loss can berealized or “locked-in” at the end of each trading period, as opposed towaiting for the final outcome of the event on which the relevantcontingent claims are based. Such lock-in can be achieved by placinghedging investments in successive trading periods as the returns change,or adjust, from period to period. In this way, profit and loss can berealized on an evolving basis (limited only by the frequency and lengthof the periods), enabling traders to achieve the same or perhaps higherfrequency of trading and hedging than available in traditional markets.

If desired, an issuer such as a corporation, investment bank,underwriter or other financial intermediary can create a security havingreturns that are driven in a comparable manner to the DBAR contingentclaims of the present invention. For example, a corporation may issue abond with returns that are linked to insurance risk. The issuer cansolicit trading and calculate the returns based on the amounts investedin contingent claims corresponding to each level or state of insurancerisks.

In a preferred embodiment of the present invention, changes in thereturn for investments in one state will affect the return oninvestments in another state in the same distribution of states for agroup of contingent claims. Thus, traders' returns will depend not onlyon the actual outcome of a real-world, observable event but also ontrading choices from among the distribution of states made by othertraders. This aspect of DBAR markets, in which returns for one state areaffected by changes in investments in another state in the samedistribution, allows for the elimination of order-crossing and dynamicmarket maker hedging. Price-discovery in preferred embodiments of thepresent invention can be supported by a one-way market (i.e., demand,not supply) for DBAR contingent claims. By structuring derivatives andinsurance trading according to DBAR principles, the high costs oftraditional order matching and principal market making market structurescan be reduced substantially. Additionally, a market implemented bysystems and methods of the present invention is especially amenable toelectronic operation over a wide network, such as the Internet.

In its preferred embodiments, the present invention mitigatesderivatives transaction costs found in traditional markets due todynamic hedging and order matching. A preferred embodiment of thepresent invention provides a system for trading contingent claimsstructured under DBAR principles, in which amounts invested in on eachstate in a group of DBAR contingent claims are reallocated fromunsuccessful investments, under defined rules, to successful investmentsafter the deduction of exchange transaction fees. In particular, theoperator of such a system or exchange provides the physical plant andelectronic infrastructure for trading to be conducted, collects andaggregates investments (or in one embodiment, first collects andaggregates investment information to determine investment amounts pertrade or order and then collects and aggregates the investment amounts),calculates the returns that result from such investments, and thenallocates to the successful investments returns that are financed by theunsuccessful investments, after deducting a transaction fee for theoperation of the system.

In preferred embodiments, where the successful investments are financedwith the losses from unsuccessful investments, returns on all trades arecorrelated and traders make investments against each other as well asassuming the risk of chance outcomes. All traders for a group of DBARcontingent claims depending on a given event become counterparties toeach other, leading to a mutualization of financial interests.Furthermore, in preferred embodiments of the present invention,projected returns prevailing at the time an investment is made may notbe the same as the final payouts or returns after the outcome of therelevant event is known.

Traditional derivatives markets by contrast, operate largely under ahouse “banking” system. In this system, the market-maker, whichtypically has the function of matching buyers and sellers, customarilyquotes a price at which an investor may buy or sell. If a given investorbuys or sells at the price, the investor's ultimate return is based uponthis price, i.e., the price at which the investor later sells or buysthe original position, along with the original price at which theposition was traded, will determine the investor's return. As themarket-maker may not be able perfectly to offset buy and sell orders atall times or may desire to maintain a degree of risk in the expectationof returns, it will frequently be subject to varying degrees of marketrisk (as well as credit risk, in some cases). In a traditionalderivatives market, market-makers which match buy and sell orderstypically rely upon actuarial advantage, bid-offer spreads, a largecapital base, and “coppering” or hedging (risk management) to minimizethe chance of bankruptcy due to such market risk exposures.

Each trader in a house banking system typically has only a singlecounterparty—the market-maker, exchange, or trading counterparty (in thecase, for example, of over-the-counter derivatives). By contrast,because a market in DBAR contingent claims may operate according toprinciples whereby unsuccessful investments finance the returns onsuccessful investments, the exchange itself is exposed to reduced riskof loss and therefore has reduced need to transact in the market tohedge itself. In preferred embodiments of DBAR contingent claims of thepresent invention, dynamic hedging or bid-offer crossing by the exchangeis generally not required, and the probability of the exchange ormarket-maker going bankrupt may be reduced essentially to zero. Such asystem distributes the risk of bankruptcy away from the exchange ormarket-maker and among all the traders in the system. The system as awhole provides a great degree of self-hedging and substantial reductionof the risk of market failure for reasons related to market risk. A DBARcontingent claim exchange or market or auction may also be“self-clearing” and require little clearing infrastructure (such asclearing agents, custodians, nostro/vostro bank accounts, and transferand register agents). A derivatives trading system or exchange or marketor auction structured according to DBAR contingent claim principlestherefore offers many advantages over current derivatives marketsgoverned by house banking principles.

The present invention also differs from electronic or parimutuel bettingsystems disclosed in the prior art (e.g., U.S. Pat. Nos. 5,873,782 and5,749,785). In betting systems or games of chance, in the absence of awager the bettor is economically indifferent to the outcome (assumingthe bettor does not own the casino or the racetrack or breed the racinghorses, for example). The difference between games of chance and eventsof economic significance is well known and understood in financialmarkets.

In summary, the present invention provides systems and methods forconducting demand-based trading. A preferred embodiment of a method ofthe present invention for conducting demand-based trading includes thesteps of (a) establishing a plurality of defined states and a pluralityof predetermined termination criteria, wherein each of the definedstates corresponds to at least one possible outcome of an event ofeconomic significance; (b) accepting investments of value units by aplurality of traders in the defined states; and (c) allocating a payoutto each investment. The allocating step is responsive to the totalnumber of value units invested in the defined states, the relativenumber of value units invested in each of the defined states, and theidentification of the defined state that occurred upon fulfillment ofall of the termination criteria.

An additional preferred embodiment of a method for conductingdemand-based trading also includes establishing, accepting, andallocating steps. The establishing step in this embodiment includesestablishing a plurality of defined states and a plurality ofpredetermined termination criteria. Each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled. The accepting stepincludes accepting investments of value units by multiple traders in thedefined states. The allocating step includes allocating a payout to eachinvestment. This allocating step is responsive to the total number ofvalue units invested in the defined states, the relative number of valueunits invested in each of the defined states, and the identification ofthe defined state that occurred upon fulfillment of all of thetermination criteria.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, the payout to each investment in each of thedefined states that did not occur upon fulfillment of all of thetermination criteria is zero, and the sum of the payouts to all of theinvestments is not greater than the value of the total number of thevalue units invested in the defined states. In a further preferredembodiment, the sum of the values of the payouts to all of theinvestments is equal to the value of all of the value units invested indefined states, less a fee.

In preferred embodiments of a method for conducting demand-basedtrading, at least one investment of value units designates a set ofdefined states and a desired return-on-investment from the designatedset of defined states. In these preferred embodiments, the allocatingstep is further responsive to the desired return-on-investment from thedesignated set of defined states.

In another preferred embodiment of a method for conducting demand-basedtrading, the method further includes the step of calculatingCapital-At-Risk for at least one investment of value units by at leastone trader. In alternative further preferred embodiments, the step ofcalculating Capital-At-Risk includes the use of the Capital-At-RiskValue-At-Risk method, the Capital-At-Risk Monte Carlo Simulation method,or the Capital-At-Risk Historical Simulation method.

In preferred embodiments of a method for conducting demand-basedtrading, the method further includes the step of calculatingCredit-Capital-At-Risk for at least one investment of value units by atleast one trader. In alternative further preferred embodiments, the stepof calculating Credit-Capital-At-Risk includes the use of theCredit-Capital-At-Risk Value-At-Risk method, the Credit-Capital-At-RiskMonte Carlo Simulation method, or the Credit-Capital-At-Risk HistoricalSimulation method.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, at least one investment of value units is amulti-state investment that designates a set of defined states. In afurther preferred embodiment, at least one multi-state investmentdesignates a set of desired returns that is responsive to the designatedset of defined states, and the allocating step is further responsive tothe set of desired returns. In a further preferred embodiment, eachdesired return of the set of desired returns is responsive to a subsetof the designated set of defined states. In an alternative preferredembodiment, the set of desired returns approximately corresponds toexpected returns from a set of defined states of a prespecifiedinvestment vehicle such as, for example, a particular call option.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, the allocating step includes the steps of (a)calculating the required number of value units of the multi-stateinvestment that designates a set of desired returns, and (b)distributing the value units of the multi-state investment thatdesignates a set of desired returns to the plurality of defined states.In a further preferred embodiment, the allocating step includes the stepof solving a set of simultaneous equations that relate traded amounts tounit payouts and payout distributions; and the calculating step and thedistributing step are responsive to the solving step.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, the solving step includes the step of fixedpoint iteration. In further preferred embodiments, the step of fixedpoint iteration includes the steps of (a) selecting an equation of theset of simultaneous equations described above, the equation having anindependent variable and at least one dependent variable; (b) assigningarbitrary values to each of the dependent variables in the selectedequation; (c) calculating the value of the independent variable in theselected equation responsive to the currently assigned values of eachthe dependent variables; (d) assigning the calculated value of theindependent variable to the independent variable; (e) designating anequation of the set of simultaneous equations as the selected equation;and (f) sequentially performing the calculating the value step, theassigning the calculated value step, and the designating an equationstep until the value of each of the variables converges.

A preferred embodiment of a method for estimating state probabilities ina demand-based trading method of the present invention includes thesteps of: (a) performing a demand-based trading method having aplurality of defined states and a plurality of predetermined terminationcriteria, wherein an investment of value units by each of a plurality oftraders is accepted in at least one of the defined states, and at leastone of these defined states corresponds to at least one possible outcomeof an event of economic significance; (b) monitoring the relative numberof value units invested in each of the defined states; and (c)estimating, responsive to the monitoring step, the probability that aselected defined state will be the defined state that occurs uponfulfillment of all of the termination criteria.

An additional preferred embodiment of a method for estimating stateprobabilities in a demand-based trading method also includes performing,monitoring, and estimating steps. The performing step includesperforming a demand-based trading method having a plurality of definedstates and a plurality of predetermined termination criteria, wherein aninvestment of value units by each of a plurality of traders is acceptedin at least one of the defined states; and wherein each of the definedstates corresponds to a possible state of a selected financial productwhen each of the termination criteria is fulfilled. The monitoring stepincludes monitoring the relative number of value units invested in eachof the defined states. The estimating step includes estimating,responsive to the monitoring step, the probability that a selecteddefined state will be the defined state that occurs upon fulfillment ofall of the termination criteria.

A preferred embodiment of a method for promoting liquidity in ademand-based trading method of the present invention includes the stepof performing a demand-based trading method having a plurality ofdefined states and a plurality of predetermined termination criteria,wherein an investment of value units by each of a plurality of tradersis accepted in at least one of the defined states and wherein anyinvestment of value units cannot be withdrawn after acceptance. Each ofthe defined states corresponds to at least one possible outcome of anevent of economic significance. A further preferred embodiment of amethod for promoting liquidity in a demand-based trading method includesthe step of hedging. The hedging step includes the hedging of a trader'sprevious investment of value units by making a new investment of valueunits in one or more of the defined states not invested in by theprevious investment.

An additional preferred embodiment of a method for promoting liquidityin a demand-based trading method includes the step of performing ademand-based trading method having a plurality of defined states and aplurality of predetermined termination criteria, wherein an investmentof value units by each of a plurality of traders is accepted in at leastone of the defined states and wherein any investment of value unitscannot be withdrawn after acceptance, and each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled. A further preferredembodiment of such a method for promoting liquidity in a demand-basedtrading method includes the step of hedging. The hedging step includesthe hedging of a trader's previous investment of value units by making anew investment of value units in one or more of the defined states notinvested in by the previous investment.

A preferred embodiment of a method for conducting quasi-continuousdemand-based trading includes the steps of: (a) establishing a pluralityof defined states and a plurality of predetermined termination criteria,wherein each of the defined states corresponds to at least one possibleoutcome of an event; (b) conducting a plurality of trading cycles,wherein each trading cycle includes the step of accepting, during apredefined trading period and prior to the fulfillment of all of thetermination criteria, an investment of value units by each of aplurality of traders in at least one of the defined states; and (c)allocating a payout to each investment. The allocating step isresponsive to the total number of the value units invested in thedefined states during each of the trading periods, the relative numberof the value units invested in each of the defined states during each ofthe trading periods, and an identification of the defined state thatoccurred upon fulfillment of all of the termination criteria. In afurther preferred embodiment of a method for conducting quasi-continuousdemand-based trading, the predefined trading periods are sequential anddo not overlap.

Another preferred embodiment of a method for conducting demand-basedtrading includes the steps of: (a) establishing a plurality of definedstates and a plurality of predetermined termination criteria, whereineach of the defined states corresponds to one possible outcome of anevent of economic significance (or a financial instrument); (b)accepting, prior to fulfillment of all of the termination criteria, aninvestment of value units by each of a plurality of traders in at leastone of the plurality of defined states, with at least one investmentdesignating a range of possible outcomes corresponding to a set ofdefined states; and (c) allocating a payout to each investment. In sucha preferred embodiment, the allocating step is responsive to the totalnumber of value units in the plurality of defined states, the relativenumber of value units invested in each of the defined states, and anidentification of the defined state that occurred upon the fulfillmentof all of the termination criteria. Also in such a preferred embodiment,the allocation is done so that substantially the same payout isallocated to each state of the set of defined states. This embodimentcontemplates, among other implementations, a market or exchange forcontingent claims of the present invention that provides—withouttraditional sellers—profit and loss scenarios comparable to thoseexpected by traders in derivative securities known as digital options,where payout is the same if the option expires anywhere in the money,and where there is no payout if the option expires out of the money.

Another preferred embodiment of the present invention provides a methodfor conducting demand-based trading including: (a) establishing aplurality of defined states and a plurality of predetermined terminationcriteria, wherein each of the defined states corresponds to one possibleoutcome of an event of economic significance (or a financialinstrument); (b) accepting, prior to fulfillment of all of thetermination criteria, a conditional investment order by a trader in atleast one of the plurality of defined states; (c) computing, prior tofulfillment of all of the termination criteria a probabilitycorresponding to each defined state; and (d) executing or withdrawing,prior to the fulfillment of all of the termination criteria, theconditional investment responsive to the computing step. In suchembodiments, the computing step is responsive to the total number ofvalue units invested in the plurality of defined states and the relativenumber of value units invested in each of the plurality of definedstates. Such embodiments contemplate, among other implementations, amarket or exchange (again without traditional sellers) in whichinvestors can make and execute conditional or limit orders, where anorder is executed or withdrawn in response to a calculation of aprobability of the occurrence of one or more of the defined states.Preferred embodiments of the system of the present invention involve theuse of electronic technologies, such as computers, computerizeddatabases and telecommunications systems, to implement methods forconducting demand-based trading of the present invention.

A preferred embodiment of a system of the present invention forconducting demand-based trading includes (a) means for accepting, priorto the fulfillment of all predetermined termination criteria,investments of value units by a plurality of traders in at least one ofa plurality of defined states, wherein each of the defined statescorresponds to at least one possible outcome of an event of economicsignificance; and (b) means for allocating a payout to each investment.This allocation is responsive to the total number of value unitsinvested in the defined states, the relative number of value unitsinvested in each of the defined states, and the identification of thedefined state that occurred upon fulfillment of all of the terminationcriteria.

An additional preferred embodiment of a system of the present inventionfor conducting demand-based trading includes (a) means for accepting,prior to the fulfillment of all predetermined termination criteria,investments of value units by a plurality of traders in at least one ofa plurality of defined states, wherein each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled; and (b) means forallocating a payout to each investment. This allocation is responsive tothe total number of value units invested in the defined states, therelative number of value units invested in each of the defined states,and the identification of the defined state that occurred uponfulfillment of all of the termination criteria.

A preferred embodiment of a demand-based trading apparatus of thepresent invention includes (a) an interface processor communicating witha plurality of traders and a market data system; and (b) a demand-basedtransaction processor, communicating with the interface processor andhaving a trade status database. The demand-based transaction processormaintains, responsive to the market data system and to a demand-basedtransaction with one of the plurality of traders, the trade statusdatabase, and processes, responsive to the trade status database, thedemand-based transaction.

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, maintaining the trade status database includes(a) establishing a contingent claim having a plurality of definedstates, a plurality of predetermined termination criteria, and at leastone trading period, wherein each of the defined states corresponds to atleast one possible outcome of an event of economic significance; (b)recording, responsive to the demand-based transaction, an investment ofvalue units by one of the plurality of traders in at least one of theplurality of defined states; (c) calculating, responsive to the totalnumber of the value units invested in the plurality of defined statesduring each trading period and responsive to the relative number of thevalue units invested in each of the plurality of defined states duringeach trading period, finalized returns at the end of each tradingperiod; and (d) determining, responsive to an identification of thedefined state that occurred upon the fulfillment of all of thetermination criteria and to the finalized returns, payouts to each ofthe plurality of traders; and processing the demand-based transactionincludes accepting, during the trading period, the investment of valueunits by one of the plurality of traders in at least one of theplurality of defined states;

In an alternative further preferred embodiment of a demand-based tradingapparatus of the present invention, maintaining the trade statusdatabase includes (a) establishing a contingent claim having a pluralityof defined states, a plurality of predetermined termination criteria,and at least one trading period, wherein each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled; (b) recording, responsiveto the demand-based transaction, an investment of value units by one ofthe plurality of traders in at least one of the plurality of definedstates; (c) calculating, responsive to the total number of the valueunits invested in the plurality of defined states during each tradingperiod and responsive to the relative number of the value units investedin each of the plurality of defined states during each trading period,finalized returns at the end of each trading period; and (d)determining, responsive to an identification of the defined state thatoccurred upon the fulfillment of all of the termination criteria and tothe finalized returns, payouts to each of the plurality of traders; andprocessing the demand-based transaction includes accepting, during thetrading period, the investment of value units by one of the plurality oftraders in at least one of the plurality of defined states;

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, maintaining the trade status database includescalculating return estimates; and processing the demand-basedtransaction includes providing, responsive to the demand-basedtransaction, the return estimates.

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, maintaining the trade status database includescalculating risk estimates; and processing the demand-based transactionincludes providing, responsive to the demand-based transaction, the riskestimates.

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, the demand-based transaction includes amulti-state investment that specifies a desired payout distribution anda set of constituent states; and maintaining the trade status databaseincludes allocating, responsive to the multi-state investment, valueunits to the set of constituent states to create the desired payoutdistribution. Such demand-based transactions may also includemulti-state investments that specify the same payout if any of adesignated set of states occurs upon fulfillment of the terminationcriteria. Other demand-based transactions executed by the demand-basedtrading apparatus of the present invention include conditionalinvestments in one or more states, where the investment is executed orwithdrawn in response to a calculation of a probability of theoccurrence of one or more states upon the fulfillment of the terminationcriteria.

In an additional embodiment, systems and methods for conductingdemand-based trading includes the steps of (a) establishing a pluralityof states, each state corresponding to at least one possible outcome ofan event of economic significance; (b) receiving an indication of adesired payout and an indication of a selected outcome, the selectedoutcome corresponding to at least one of the plurality of states; and(c) determining an investment amount as a function of the selectedoutcome, the desired payout and a total amount invested in the pluralityof states.

In another additional embodiment, systems and methods for conductingdemand-based trading includes the steps of (a) establishing a pluralityof states, each state corresponding to at least one possible outcome ofan event (whether or not such event is an economic event); (b) receivingan indication of a desired payout and an indication of a selectedoutcome, the selected outcome corresponding to at least one of theplurality of states; and (c) determining an investment amount as afunction of the selected outcome, the desired payout and a total amountinvested in the plurality of states.

In another additional embodiment, systems and methods for conductingdemand-based trading includes the steps of (a) establishing a pluralityof states, each state corresponding to at least one possible outcome ofan event of economic significance; (b) receiving an indication of aninvestment amount and a selected outcome, the selected outcomecorresponding to at least one of the plurality of states; and (c)determining a payout as a function of the investment amount, theselected outcome, a total amount invested in the plurality of states,and an identification of at least one state corresponding to an observedoutcome of the event.

In another additional embodiment, systems and methods for conductingdemand-based trading include the steps of: (a) receiving an indicationof one or more parameters of a financial product or derivativesstrategy; and (b) determining one or more of a selected outcome, adesired payout, an investment amount, and a limit on the investmentamount for each contingent claim in a set of one or more contingentclaims as a function of the one or more financial product or derivativesstrategy parameters.

In another additional embodiment, systems and methods for conductingdemand-based trading include the steps of: (a) receiving an indicationof one or more parameters of a financial product or derivativesstrategy; and (b) determining an investment amount and a selectedoutcome for each contingent claim in a set of one or more contingentclaims as a function of the one or more financial product or derivativesstrategy parameters.

In another additional embodiment, a demand-enabled financial product fortrading in a demand-based auction includes a set of one or morecontingent claims, the set approximating or replicating a financialproduct or derivatives strategy, each contingent claim in the set havingan investment amount and a selected outcome, each investment amountbeing dependent upon one or more parameters of a financial product orderivatives strategy and a total amount invested in the auction.

In another additional embodiment, methods for conducting demand-basedtrading on at least one event includes the steps of: (a) determining oneor more parameters of a contingent claim, in a replication set of one ormore contingent claims, as a function of one or more parameters of aderivatives strategy and an outcome of the event; and (b) determining aninvestment amount for a contingent claim in the replication set as afunction of one or more parameters of the derivatives strategy and anoutcome of the event.

In another additional embodiment, methods for conducting demand basedtrading include the steps of: enabling one or more derivativesstrategies and/or financial products to be traded in a demand-basedauction; and offering and/or trading one or more of the enabledderivatives strategies and enabled financial products to customers.

In another additional embodiment, methods for conducting derivativestrading include the steps of: receiving an indication of one or moreparameters of a derivatives strategy on one or more events of economicsignificance; and determining one or more parameters of each digital ina replication set made up of one or more digitals as a function of oneor more parameters of the derivatives strategy.

In another additional embodiment, methods for trading contingent claimsin a demand-based auction, includes the step of approximating orreplicating a contingent claim with a set of demand-based claims. Theset of demand-based claims includes at least one vanilla option, thusdefining a vanilla replicating basis.

In another additional embodiment, methods for trading contingent claimsin a demand-based auction on an event, includes the step of: determininga value of a contingent claim as a function of a demand-based valuationof each vanilla option in a replication set for the contingent claim.The replication set includes at least one vanilla option, thus defininganother vanilla replicating basis.

In another additional embodiment, methods for conducting a demand-basedauction on an event, includes the steps of: establishing a plurality ofstrikes for the auction, each strike corresponding to a possible outcomeof the event; establishing a plurality of replicating claims for theauction, one or more replicating claims striking at each strike in theplurality of strikes; replicating a contingent claim with a replicationset including one or more of the replicating claims; and determining theprice and/or payout of the contingent claim as a function of ademand-based valuation of each of the replicating claims in thereplication set.

In another additional embodiment, methods for processing a customerorder for one or more derivatives strategies, in a demand-based auctionon an event, where the auction includes one or more customer orders aredescribed as including the steps of: establishing strikes for theauction, each one of the strikes corresponding to a possible outcome ofthe event; establishing replicating claims for the auction, one or morereplicating claims striking at each strike in the auction; replicatingeach derivatives strategy in the customer order with a replication setincluding one or more of the replicating claims in the auction; anddetermining a premium for the customer order by engaging in ademand-based valuation of each one of the replicating claims in thereplication set for each one of the derivatives strategies in thecustomer order.

In another additional embodiment, a method for investing in ademand-based auction on an event, includes the steps of: providing anindication of one or more selected strikes and a payout profile for oneor more derivatives strategies, each of the selected strikescorresponding to a selected outcome of the event, and each of theselected strikes being selected from a plurality of strikes establishedfor the auction, each of the strikes corresponding to a possible outcomeof the event; receiving an indication of a price for each of thederivatives strategies, the price being determined by engaging in ademand-based valuation of a replication set replicating the derivativesstrategy, the replication set including one or more replicating claimsfrom a plurality of replicating claims established for the auction, atleast one of each of the replicating claims in the auction striking atone of the strikes.

In another additional embodiment, a computer system for processing acustomer order for one or more derivatives strategy, in a demand-basedauction on an event, the auction including one or more customer orders,the computer system including one or more processors that are configuredto: establish strikes for the auction, each one of the strikescorresponding to a possible outcome of the event; establish replicatingclaims for the auction, one or more replicating claims striking at eachone of the strikes; and replicate each of the derivatives strategies inthe customer order with a replication set including one or more of thereplicating claims in the auction; and determine a premium for thecustomer order by engaging in a demand-based valuation of each one ofthe replicating claims in the replication set for each one of thederivatives strategies in the customer order.

In another additional embodiment, a computer system for placing an orderto invest in a demand-based auction on an event, the order including oneor more derivatives strategies, the computer system including one ormore processors configured to: provide an indication of one or moreselected strikes and a payout profile for each derivatives strategy,each selected strike corresponding to a selected outcome of the event,and each selected strike being selected from a plurality of strikesestablished for the auction, each of the strikes corresponding to apossible outcome of the event; receive an indication of a premium forthe order, the premium of the order being determined by engaging in ademand-based valuation of a replication set replicating each derivativesstrategy in the order, the replication set including one or morereplicating claims from a plurality of replicating claims establishedfor the auction, with one or more of the replicating claims in theauction striking at each of the strikes.

In another additional embodiment, a method for executing a tradeincludes the steps of: receiving a request for an order, the requestindicating one or more selected strikes and a payout profile for one ormore derivatives strategies in the order, each selected strikecorresponding to a selected outcome of the event, and each selectedstrike being selected from a plurality of strikes established for theauction, each of the strikes corresponding to a possible outcome of theevent; providing an indication of a premium for the order, the premiumbeing determined by engaging in a demand-based valuation of areplication set replicating each derivatives strategy in the order, thereplication set including one or more replicating claims from aplurality of replicating claims established for the auction, one or moreof each of the replicating claims in the auction striking at each of thestrikes; and receiving an indication of a decision to place the orderfor the determined premium.

In another additional embodiment, a method for providing financialadvice, includes the steps of: providing a person with advice aboutinvesting in one or more of a type of derivatives strategy in ademand-based auction, an order for the one or more derivativesstrategies indicating one or more selected strikes and a payout profilefor the derivatives strategy, each selected strike corresponding to aselected outcome of the event, and each selected strike being selectedfrom a plurality of strikes established for the auction, each of thestrikes corresponding to a possible outcome of the event, wherein thepremium for the order is determined by engaging in a demand-basedvaluation of a replication set replicating each of the derivativesstrategies in the order, the replication set including at least onereplicating claim from a plurality of replicating claims established forthe auction, one or more of the replicating claims in the auctionstriking at one of the strikes.

In another additional embodiment, a method of hedging, includes thesteps of: determining an investment risk in one or more investments; andoffsetting the investment risk by taking a position in one or morederivatives strategies in a demand-based auction with an opposing risk,an order for the one or more derivatives strategies indicating one ormore selected strikes and a payout profile for the derivatives strategyin the order, each selected strike corresponding to a selected outcomeof the event, and each selected strike being selected from a pluralityof strikes established for the auction, each of the strikescorresponding to a possible outcome of the event, wherein the premiumfor the order is determined by engaging in a demand-based valuation of areplication set replicating each of the derivatives strategies in theorder, the replication set including at least one replicating claim froma plurality of replicating claims established for the auction, one ormore of each of the replicating claims in the auction striking at one ofthe strikes.

In another additional embodiment, a method of speculating, includes thesteps of: determining an investment risk in at least one investment; andincreasing the investment risk by taking a position in one or morederivatives strategies in a demand-based auction with a similar risk, anorder for the one or more derivatives strategies. The order specifiesone or more selected strikes and a payout profile for the derivativesstrategy, and can also specify a requested number of the derivativesstrategy. Each selected strike corresponds to a selected outcome of theevent, each selected strike is selected from a plurality of strikesestablished for the auction, and each of the strikes corresponds to apossible outcome of the event. The premium for the order is determinedby engaging in a demand-based valuation of a replication set replicatingeach of the derivatives strategies in the order, the replication setincluding one or more replicating claims from a plurality of replicatingclaims established for the auction, one or more of the replicatingclaims in the auction striking at each one of the strikes.

In another additional embodiment, a computer program product capable ofprocessing a customer order including one or more derivativesstrategies, in a demand-based auction including one or more customerorders, the computer program product including a computer usable mediumhaving computer readable program code embodied in the medium for causinga computer to: establish strikes for the auction, each one of thestrikes corresponding to a possible outcome of the event; establishreplicating claims for the auction, one or more of the replicatingclaims striking at one of the strikes; and replicate each derivativesstrategy in the customer order with a replication set including at leastone of the replicating claims in the auction; and determine a premiumfor the customer order by engaging in a demand-based valuation of eachof the replicating claims in the replication set for each of thederivatives strategies in the customer order.

In another additional embodiment, an article of manufacture comprisingan information storage medium encoded with a computer-readable datastructure adapted for use in placing a customer order in a demand-basedauction over the Internet, the auction including at least one customerorder, said data structure including: at least one data field withinformation identifying one or more selected strikes and a payoutprofile for each of the derivatives strategies in the customer order,each selected strike corresponding to a selected outcome of the event,and each selected strike being selected from a plurality of strikesestablished for the auction, each strike in the auction corresponding toa possible outcome of the event; and one or more data fields withinformation identifying a premium for the order, the premium beingdetermined as a result of a demand-based valuation of a replication setreplicating each of the derivatives strategies in the order, thereplication set including at least one replicating claim from aplurality of replicating claims established for the auction, one or moreof each of the replicating claims in the auction striking at one of thestrikes.

In another additional embodiment, a derivatives strategy for ademand-based market, includes: a first designation of at least oneselected strike for the derivatives strategy, each selected strike beingselected from a plurality of strikes established for auction, eachstrike in the auction corresponding to a possible outcome of the event;a second designation of a payout profile for the derivatives strategy;and a price for the derivatives strategy, the price being determined byengaging in a demand-based valuation of a replication set replicatingthe first designation and the second designation of the derivativesstrategy, the replication set including one or more replicating claimsfrom a plurality of replicating claims established for the auction, oneor more of the replicating claims in the auction striking at each strikein the auction.

In another additional embodiment, an investment vehicle for ademand-based auction, includes: a demand-based derivatives strategyproviding investment capital to the auction, an amount of the providedinvestment capital being dependent upon a demand-based valuation of areplication set replicating the derivatives strategy, the replicatingset including one or more of the replicating claims from a plurality ofreplicating claims established for the auction, one or more of thereplicating claims in the auction striking at each one of the strikes inthe auction.

In another additional embodiment, an article of manufacture comprising apropagated signal adapted for use in the performance of a method fortrading a customer order including at least one of a derivativesstrategy, in a demand-based auction including one or more customerorders, wherein the method includes the steps of: establishing strikesfor the auction, each one of the strikes corresponding to a possibleoutcome of the event; establishing replicating claims for the auction,one or more of the replicating claims striking at one of the strikes;replicating each one of the derivatives strategies in the customer orderwith a replication set including one or more of the replicating claimsin the auction; and determining a premium for the customer order byengaging in a demand-based valuation of each one of the replicatingclaims in the replication set for the derivatives strategy in thecustomer order; wherein the propagated signal is encoded withmachine-readable information relating to the trade.

In another additional embodiment, a computer system for conductingdemand-based auctions on an event, includes one or more user interfaceprocessors, a database unit, an auction processor and a calculationengine. The one or more interface processors are configured tocommunicate with a plurality of terminals which are adapted to enterdemand-based order data for an auction. The database unit is configuredto maintain an auction information database. The auction processor isconfigured to process at least one demand-based auction and tocommunicate with the user interface processor and the database unit,wherein the auction processor is configured to generate auctiontransaction data based on auction order data received from the userinterface processor and to send the auction transaction data for storingto the database unit, and wherein the auction processor is furtherconfigured to establish a plurality of strikes for the auction, eachstrike corresponding to a possible outcome of the event, to establish aplurality of replicating claims for the auction, at least onereplicating claim striking at a strike in the plurality of strikes, toreplicate a contingent claim with a replication set including at leastone of the plurality of replicating claims, and to send the replicationset for storing to the database unit. The calculation engine isconfigured to determine at least one of an equilibrium price and apayout for the contingent claim as a function of a demand-basedvaluation of each of the replicating claims in the replication setstored in the database unit.

An object of the present invention is to provide systems and methods tosupport and facilitate a market structure for contingent claims relatedto observable events of economic significance, which includes one ormore of the following advantages, in addition to those described above:

-   1. ready implementation and support using electronic computing and    networking technologies;-   2. reduction or elimination of the need to match bids to buy with    offers to sell in order to create a market for derivatives;-   3. reduction or elimination of the need for a derivatives    intermediary to match bids and offers;-   4. mathematical and consistent calculation of returns based on    demand for contingent claims;-   5. increased liquidity and liquidity incentives;-   6. statistical diversification of credit risk through the    mutualization of multiple derivatives counterparties;-   7. improved scalability by reducing the traditional linkage between    the method of pricing for contingent claims and the quantity of the    underlying claims available for investment;-   8. increased price transparency;-   9. improved efficiency of information aggregation mechanisms;-   10. reduction of event risk, such as the risk of discontinuous    market events such as crashes;-   11. opportunities for binding offers of liquidity to the market;-   12. reduced incentives for strategic behavior by traders;-   13. increased market for contingent claims;-   14. improved price discovery;-   15. improved self-consistency;-   16. reduced influence by market makers;-   17. ability to accommodate virtually unlimited demand;-   18. ability to isolate risk exposures;-   19. increased trading precision, transaction certainty and    flexibility;-   20. ability to create valuable new markets with a sustainable    competitive advantage;-   21. new source of fee revenue without putting capital at risk; and-   22. increased capital efficiency.

A further object of the present invention is to provide systems andmethods for the electronic exchange of contingent claims related toobservable events of economic significance, which includes one or moreof the following advantages:

-   1. reduced transaction costs, including settlement and clearing    costs, associated with derivatives transactions and insurable    claims;-   2. reduced dependence on complicated valuation models for trading    and risk management of derivatives;-   3. reduced need for an exchange or market maker to manage market    risk by hedging;-   4. increased availability to traders of accurate and up-to-date    information on the trading of contingent claims, including    information regarding the aggregate amounts invested across all    states of events of economic significance, and including over    varying time periods;-   5. reduced exposure of the exchange to credit risk;-   6. increased availability of information on credit risk and market    risk borne by traders of contingent claims;-   7. increased availability of information on marginal returns from    trades and investments that can be displayed instantaneously after    the returns adjust during a trading period;-   8. reduced need for a derivatives intermediary or exchange to match    bids and offers;-   9. increased ability to customize demand-based adjustable return    (DBAR) payouts to permit replication of traditional financial    products and their derivatives;-   10. comparability of profit and loss scenarios to those expected by    traders for purchases and sales of digital options and other    derivatives, without conventional sellers;-   11. increased data generation; and-   12. reduced exposure of the exchange to market risk.

Other additional embodiments include features for an enhanced parimutuelwagering system and method, which are described in further detail inChapter 15 below. In one of such other additional embodiment, a methodfor conducting enhanced parimutuel wagering includes the steps of:enabling at least two types of bets to be offered in the same or commonbetting pool; and offering at least one of the two types of enabled betsto bettors. The different types of bets (e.g., trifecta bets, finishbet, show bets, as described in further detail in Chapter 15 below) canbe processed in the same betting pool, for example, by configuringequivalent combinations of fundamental bets for each type of bet, andperforming a demand-based valuation of each of the fundamental bets inthe betting pool.

In another additional embodiment, a method for conducting enhancedparimutuel wagering, includes the steps of: establishing a plurality offundamental outcomes; receiving an indication of limit odds, a premiumand a selected outcome; and determining the payout as a function of theselected outcome, the limit odds, the premium and the total amountwagered in the plurality of fundamental outcomes. Each fundamentaloutcome corresponds to one or more possible outcomes of an underlyingevent (e.g., a wagering event such as a lottery, a sporting event or ahorse or dog race), and the selected outcome corresponds to one or morefundamental outcomes.

In another additional embodiment, a method for conducting enhancedparimutuel wagering includes the steps of: establishing a plurality offundamental outcomes in a betting pool; receiving an indication of limitodds, a premium, and a selected outcome; and determining final odds as afunction of the selected outcome, the limit odds, the premium and atotal amount wagered in the plurality of fundamental outcomes in thebetting pool. Each fundamental outcome corresponds to one or morepossible outcomes of an underlying event, and the selected outcomecorresponds to one or more of the fundamental outcomes.

In another additional embodiment, a method for conducting enhancedparimutuel wagering, includes the steps of: establishing a plurality offundamental outcomes; receiving an indication of limit odds, a desiredpayout, and a selected outcome; and determining a premium as a functionof the selected outcome, the limit odds, the desired payout and a totalamount wagered in the plurality of fundamental outcomes. Eachfundamental outcome corresponds to one or more possible outcomes of anunderlying event; and the selected outcome corresponds to one or more ofthe fundamental outcomes.

In another additional embodiment, a method for conducting enhancedparimutuel wagering, includes the steps of: establishing a plurality offundamental outcomes in a betting pool; receiving an indication of limitodds, a desired payout, and a selected outcome; and determining finalodds as a function of the selected outcome, the limit odds, the desiredpayout and a total amount wagered in the plurality of fundamentaloutcomes in the betting pool. Each fundamental outcome corresponds toone or more possible outcomes of an underlying event, and the selectedoutcome corresponds to one or more of the fundamental outcomes.

In another additional embodiment, a method for processing a bet in anenhanced parimutuel betting pool on an underlying event, the bettingpool including one or more bets, includes the steps of: establishingfundamental outcomes in a betting pool; establishing fundamental bets oneach fundamental outcome in the betting pool; configuring an equivalentcombination of fundamental bets for the bet as a function of a selectedoutcome for the bet; and determining at least one of a premium and apayout for the wager as a function of a demand-based valuation of eachfundamental bet in the equivalent combination for the wager. Each one ofthe fundamental outcomes corresponds to one or more possible outcomes ofthe underlying event, and the selected outcome for the bet correspondsto one or more of the fundamental outcomes.

In another additional embodiment, a method for processing a bet in anenhanced parimutuel betting pool on an underlying event, the bettingpool including one or more wagers, includes the steps of: establishingfundamental outcomes in a betting pool; establishing fundamental bets oneach fundamental outcome in the betting pool; configuring an equivalentcombination of fundamental bets for the wager as a function of aselected outcome for the wager; and determining a price for eachfundamental bet as a function of a price of each of the otherfundamental bets in the betting pool, a total filled amount in eachfundamental outcome, and a total amount wagered in the plurality offundamental outcomes in the betting pool. Each one of the fundamentaloutcomes corresponds to one or more possible outcomes of the underlyingevent, and the selected outcome for the wager corresponds to one or moreof the fundamental outcomes.

In another additional embodiment, a method for betting on an underlyingevent, includes the steps of: providing an indication of limit odds, arequested premium, and a selected outcome, for a bet on a selectedoutcome of the underlying event; and receiving an indication of finalodds for the bet. The selected outcome corresponds to one or morefundamental outcomes in the betting pool, and each of the fundamentaloutcomes corresponds to one or more possible outcomes of the underlyingevent. The final odds are determined by engaging in a demand-basedvaluation of an equivalent combination of fundamental bets. Theequivalent combination includes one or more fundamental bets. Eachfundamental bet in the betting pool betting on a respective fundamentaloutcome. The equivalent combination of fundamental bets are configuredas a function of the selected outcome for the bet.

In another additional embodiment, a method for betting on an underlyingevent, includes the steps of: providing an indication of limit odds, adesired payout, and a selected outcome, for a bet on a selected outcomeof an underlying event; and receiving an indication of final odds forthe bet. The selected outcome corresponds to one or more fundamentaloutcomes, each of the fundamental outcomes in the betting poolcorresponding to one or more possible outcomes of the underlying event.The final odds are determined by engaging in a demand-based valuation ofan equivalent combination of fundamental bets. The equivalentcombination includes one or more of the fundamental bets, eachfundamental bet betting on a respective fundamental outcome. Theequivalent combination is configured as a function of the selectedoutcome for the bet.

In another additional embodiment, a vehicle for betting in an enhancedparimutuel betting pool, includes: a wager betting a filled premiumamount on a selected outcome of an underlying event, the wager includingan indication of limit odds on the selected outcome, and one of arequested premium and a desired payout on the selected outcome. Thefilled premium is determined as a function of the one of the requestedpremium and the desired payout and a comparison of the limit odds withfinal odds on the wager. The final odds are determined by engaging in ademand-based valuation of each of the fundamental bets in a combinationof fundamental bets equivalent to the wager. The combination includesone or more fundamental bets from the plurality of fundamental betsestablished for the betting pool. Each fundamental bet in the bettingpool betting on a fundamental outcome of the underlying event. Eachfundamental outcome corresponds to one or more of the possible outcomesof the underlying event. The selected outcome corresponds to one or moreof the fundamental outcomes. The combination of fundamental bets isconfigured as a function of the selected outcome of the wager.

In another additional embodiment, a computer system for conducting abetting pool on an underlying event, includes one or more processors,that is configured to: establish fundamental outcomes for the underlyingevent; establish fundamental bets for the underlying event; receive anindication of limit odds, one of a requested premium and a desiredpayout, and a selected outcome, for a wager on a selected outcome of theunderlying event; configure an equivalent combination of fundamentalbets for the wager as a function of the selected outcome of the wager;and determine final odds for the wager by engaging in a demand-basedvaluation of the fundamental bets in the equivalent combination. Each ofthe fundamental outcomes corresponds to one or more possible outcomes ofthe event. Each fundamental bet bets on a respective fundamental outcomein the betting pool. The selected outcome corresponds to one or more ofthe fundamental outcomes in the betting pool, and each fundamentaloutcome corresponds to one or more possible outcomes of the underlyingevent.

In another additional embodiment, a computer system, for placing a betin a betting pool on an underlying event, includes one or moreprocessors configured to: provide an indication of limit odds, one of adesired payout and a requested premium, and a selected outcome, for abet on a selected outcome of an underlying event; and receive anindication of final odds for the bet. The selected outcome correspondsto one or more of a plurality of fundamental outcomes in the bettingpool, with each fundamental outcome corresponding to one or morepossible outcomes of the underlying event. The final odds are determinedby having the processors engage in a demand-based valuation of anequivalent combination of fundamental bets for the bet. The equivalentcombination includes one or more fundamental bets, each of which bet ona respective fundamental outcome. The equivalent combination isconfigured as a function of the selected outcome for the bet.

In another additional embodiment, a computer program product is capableof processing a wager in a betting pool including at least one wager.The computer program product includes a computer usable medium havingcomputer readable program code embodied in the medium for causing acomputer or a system to: establish fundamental outcomes for theunderlying event; establish fundamental bets for the underlying event;receive an indication of limit odds, one of a requested premium and adesired payout, and a selected outcome on the underlying event, for awager on the selected outcome; configure an equivalent combination offundamental bets for the wager as a function of the selected outcome ofthe wager; and determine final odds for the wager by engaging in ademand-based valuation of the fundamental bets in the equivalentcombination. Each of the fundamental outcomes corresponds to one or moreof the possible outcomes of the underlying event. Each fundamental betbets on a respective fundamental outcome. The selected outcomecorresponds to one or more of a plurality of fundamental outcomes, eachfundamental outcome corresponding to a possible outcome of theunderlying event.

In another additional embodiment, an article of manufacture includes: aninformation storage medium encoded with a computer-readable datastructure adapted for placing a wager over the Internet in a bettingpool on an underlying event, the betting pool includes one or morewagers, said data structure comprising: one or more data fields withinformation identifying at least one selected outcome of an underlyingevent, limit odds, and one of a requested premium and a desired payoutfor the wager; and one or more data fields with information identifyingfinal odds for the wager. The final odds are determined as a result of ademand-based valuation of fundamental bets in a combination offundamental bets equivalent to the wager configured as a function of theselected outcome. The combination includes one or more of thefundamental bets established for the betting pool. Each fundamental betbets on a fundamental outcome of the underlying event. Each fundamentaloutcome corresponds to at least one possible outcome of the event, andthe selected outcome corresponds to one or more of the fundamentaloutcomes.

In another additional embodiment, an article of manufacture comprising apropagated signal adapted for use in the performance of a method forconducting a betting pool on an underlying event, the betting poolincludes one or more wagers. The signal is encoded with machine-readableinformation relating to the wager. The method includes the steps of:establishing fundamental outcomes for the underlying event; establishingfundamental bets for the underlying event; receiving an indication oflimit odds, a requested premium and/or a desired payout, and a selectedoutcome, for a wager on a selected outcome on the underlying event;configuring an equivalent combination of fundamental bets for the wageras a function of the selected outcome of the wager; and determiningfinal odds for the wager by engaging in a demand-based valuation of thefundamental bets in the equivalent combination. Each fundamental outcomecorresponds to one or more possible outcomes of the event. Eachfundamental bet bets on a respective fundamental outcome. The selectedoutcome corresponds to one or more of a plurality of fundamentaloutcomes. Each of the fundamental outcomes corresponds to one or more ofthe possible outcomes of the event. Each fundamental bet bets on arespective fundamental outcome. The selected outcome corresponds to oneor more of the plurality of fundamental outcomes. Each fundamentaloutcome corresponds to one or more possible outcomes of the underlyingevent.

Additional objects and advantages of the various embodiments of theinvention are set forth in part in the description which follows, and inpart are obvious from the description, or may be learned by practice ofthe invention. The objects and advantages of the invention may also berealized and attained by means of the instrumentalities, systems,methods and steps set forth in the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and from a part ofthe specification, illustrate embodiments of the present invention and,together with the description, serve to explain the principles of theinvention.

FIG. 1 is a schematic view of various forms of telecommunicationsbetween DBAR trader clients and a preferred embodiment of a DBARcontingent claims exchange implementing the present invention.

FIG. 2 is a schematic view of a central controller of a preferredembodiment of a DBAR contingent claims exchange network architectureimplementing the present invention.

FIG. 3 is a schematic depiction of the trading process on a preferredembodiment of a DBAR contingent claims exchange.

FIG. 4 depicts data storage devices of a preferred embodiment of a DBARcontingent claims exchange.

FIG. 5 is a flow diagram illustrating the processes of a preferredembodiment of DBAR contingent claims exchange in executing a DBAR rangederivatives investment.

FIG. 6 is an illustrative HTML interface page of a preferred embodimentof a DBAR contingent claims exchange.

FIG. 7 is a schematic view of market data flow to a preferred embodimentof a DBAR contingent claims exchange.

FIG. 8 is an illustrative graph of the implied liquidity effects for agroup of DBAR contingent claims.

FIG. 9 a is a schematic representation of a traditional interest rateswap transaction.

FIG. 9 b is a schematic of investor relationships for an illustrativegroup of DBAR contingent claims.

FIG. 9 c shows a tabulation of credit ratings and margin trades for eachinvestor in to an illustrative group of DBAR contingent claims.

FIG. 10 is a schematic view of a feedback process for a preferredembodiment of DBAR contingent claims exchange.

FIG. 11 depicts illustrative DBAR data structures for use in a preferredembodiment of a Demand-Based Adjustable Return Digital Options Exchangeof the present invention.

FIG. 12 depicts a preferred embodiment of a method for processing limitand market orders in a Demand-Based Adjustable Return Digital OptionsExchange of the present invention.

FIG. 13 depicts a preferred embodiment of a method for calculating amultistate composite equilibrium in a Demand-Based Adjustable ReturnDigital Options Exchange of the present invention.

FIG. 14 depicts a preferred embodiment of a method for calculating amultistate profile equilibrium in a Demand-Based Adjustable ReturnDigital Options Exchange of the present invention.

FIG. 15 depicts a preferred embodiment of a method for converting “sale”orders to buy orders in a Demand-Based Adjustable Return Digital OptionsExchange of the present invention.

FIG. 16: depicts a preferred embodiment of a method for adjustingimplied probabilities for demand-based adjustable return contingentclaims to account for transaction or exchange fees in a Demand-BasedAdjustable Return Digital Options Exchange of the present invention.

FIG. 17 depicts a preferred embodiment of a method for filling andremoving lots of limit orders in a Demand-Based Adjustable ReturnDigital Options Exchange of the present invention.

FIG. 18 depicts a preferred embodiment of a method of payoutdistribution and fee collection in a Demand-Based Adjustable ReturnDigital Options Exchange of the present invention.

FIG. 19 depicts illustrative DBAR data structures used in anotherembodiment of a Demand-Based Adjustable Return Digital Options Exchangeof the present invention.

FIG. 20 depicts another embodiment of a method for processing limit andmarket orders in another embodiment of a Demand-Based Adjustable ReturnDigital Options Exchange of the present invention.

FIG. 21 depicts an upward shift in the earnings expectations curve whichcan be protected by trading digital options and other contingent claimson earnings in successive quarters according to the embodiments of thepresent invention.

FIG. 22 depicts a network implementation of a demand-based market orauction according to the embodiments of the present invention.

FIG. 23 depicts cash flows for each participant trading aprinciple-protected ECI-linked FRN.

FIG. 24 depicts an example time line for a demand-based market tradingDBAR-enabled FRNs or swaps according to the embodiments of the presentinvention.

FIG. 25 depicts an example of an embodiment of a demand-based market orauction with digital options and DBAR-enabled products.

FIG. 26 depicts an example of an embodiment of a demand-based market orauction with replicated derivatives strategies, digital options andother DBAR-enabled products and derivatives.

FIGS. 27A, 27B and 27C depict an example of an embodiment replicating avanilla call for a demand-based market or auction with a strike of −325.

FIGS. 28A, 28B and 28C depict an example of an embodiment replicating acall spread for a demand-based market or auction with strikes −375 and−225.

FIG. 29 depicts an example of an embodiment of a demand-based market orauction with derivatives strategies, structured instruments and otherproducts that are DBAR-enabled by replicating them into a vanillareplicating basis.

FIG. 30 illustrates the components of a digital replicating basis for anexample embodiment in which derivatives strategies are DBAR-enabled byreplicating them into the digital replicating basis.

FIG. 31 illustrates the components of the vanilla replicating basisreferenced in FIG. 29.

FIGS. 32 to 68 illustrates a DBAR System Architecture that implementsthe example embodiment depicted in FIGS. 29 and 31.

FIG. 69 shows a dependence of whether a fixed point iteration willconverge on the value of the first derivative of a function g(x) in theneighborhood of the fixed point.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

This Detailed Description of Preferred Embodiments is organized intosixteen sections. The first section provides an overview of systems andmethods for trading or investing in groups of DBAR contingent claims.The second section describes in detail some of the important features ofsystems and methods for trading or investing in groups of DBARcontingent claims. The third section of this Detailed Description ofPreferred Embodiments provides detailed descriptions of two preferredembodiments of the present invention: investments in a group of DBARcontingent claims, and investments in a portfolio of groups of suchclaims. The fourth section discusses methods for calculating risksattendant on investments in groups and portfolios of groups of DBARcontingent claims. The fifth section of this Detailed Descriptionaddresses liquidity and price/quantity relationships in preferredembodiments of systems and methods of the present invention. The sixthsection provides a detailed description of a DBAR Digital OptionsExchange. The seventh section provides a detailed description of anotherembodiment of a DBAR Digital Options Exchange. The eighth sectionpresents a network implementation of this DBAR Digital Options Exchange.The ninth section presents a structured instrument implementation of ademand-based market or auction. The tenth section presents systems andmethods for replicating derivatives strategies using contingent claimssuch as digitals or digital options, and trading such replicatedderivatives strategies in a demand-based market. The eleventh sectionpresents systems and methods for replicating derivatives strategies andother contingent claims (e.g., structured instruments), into a vanillareplicating basis (a basis including vanilla replicating claims, andsometimes also digital replicating claims), and trading such replicatedderivatives strategies in a demand-based market or auction, pricing suchderivatives strategies in the vanilla replicating basis. The twelfthsection presents a detailed description of FIGS. 1 to 28 accompanyingthis specification. The thirteenth section presents a description of theDBAR system architecture, including additional detailed descriptions offigures accompanying the specification, with particular detail directedto the embodiments described in the eleventh section, and as illustratedin FIGS. 32 to 68. The fourteenth section of the Detailed Descriptiondiscusses some of the salient advantages of the methods and systems ofthe present invention. The fifteenth section presents enhancedparimutuel wagering systems and methods. The sixteenth section is aTechnical Appendix providing additional information on the multistateallocation method of the present invention. The last section is aconclusion of the Detailed Description. More specifically, this DetailedDescription of the Preferred Embodiments is organized as follows:

1 Overview: Exchanges and Markets for DBAR Contingent claims

-   -   1.1 Exchange Design    -   1.2 Market Operation    -   1.3 Network Implementation

2 Features of DBAR Contingent claims

-   -   2.1 DBAR Contingent Claim Notation    -   2.2 Units of Investment and Payouts    -   2.3 Canonical Demand Reallocation Functions    -   2.4 Computing Investment Amounts to Achieve Desired Payouts    -   2.5 A Canonical DRF Example    -   2.6 Interest Considerations    -   2.7 Returns and Probabilities    -   2.8 Computations When Invested Amounts are Large

3 Examples of Groups of DBAR Contingent claims

-   -   3.1 DBAR Range Derivatives    -   3.2 DBAR Portfolios

4 Risk Calculations in Groups of DBAR Contingent claims

-   -   4.1 Market Risk        -   4.1.1 Capital-At-Risk Determinations        -   4.1.2 Capital-At-Risk Determinations Using Monte Carlo            Simulation Techniques        -   4.1.3 Capital-At-Risk Determinations Using Historical            Simulation Techniques    -   4.2 Credit Risk        -   4.2.1 Credit-Capital-At-Risk Determinations        -   4.2.2 Credit-Capital-At-Risk Determinations using Monte            Carlo Simulation Techniques        -   4.2.3 Credit-Capital-At-Risk Historical Simulation            Techniques Liquidity and Price/Quantity Relationships

6 DBAR Digital Options Exchange

-   -   6.1 Representation of Digital Options as DBAR Contingent claims    -   6.2 Construction of Digital Options Using DBAR Methods and        Systems    -   6.3 Digital Option Spreads    -   6.4 Digital Option Strips    -   6.5 Multistate Allocation Algorithm for Replicating “Sell”        Trades    -   6.6 Clearing and Settlement    -   6.7 Contract Initialization    -   6.8 Conditional Investments, or Limit Orders    -   6.9 Sensitivity Analysis and Depth of Limit Order Book

6.10 Networking of DBAR Digital Options Exchanges

7 DBAR DOE: Another Embodiment

-   -   7.1 Special Notation    -   7.2 Elements of Example DBAR DOE Embodiment    -   7.3 Mathematical Principles    -   7.4 Equilibrium Algorithm    -   7.5 Sell Orders    -   7.6 Arbitrary Payout Options    -   7.7 Limit Order Book Optimization    -   7.8 Transaction Fees    -   7.9 An Embodiment of the Algorithm to Solve the Limit Order        BookOptimization    -   7.10 Limit Order Book Display    -   7.11 Unique Price Equilibrium Proof

8 Network Implementation.

9 Structured Instrument Trading

-   -   9.1 Overview: Customer Oriented DBAR-enabled Products    -   9.2 Overview: FRNs and swaps    -   9.3 Parameters: FRNs and swaps vs. digital options    -   9.4 Mechanics: DBAR-enabling FRNs and swaps    -   9.5 Example: Mapping FRNs into Digital Option Space    -   9.6 Conclusion

10 Replicating Derivatives Strategies Using Digital Options

-   -   10.1 The General Approach to Replicating Derivatives Strategies        With Digital Options    -   10.2 Application of General Results to Special Cases    -   10.3 Estimating the Distribution of the Underlying U    -   10.4 Replication P&L for a Set of Orders    -   Appendix 10A: Notation Used in Section 10    -   Appendix 10B: The General Replication Theorem    -   Appendix 10C: Derivations from Section 10.3

11 Replicating and Pricing Derivatives Strategies using Vanilla Options

-   -   11.1 Replicating Derivatives Strategies Using Digital Options    -   11.2 Replicating Claims Using a Vanilla Replicating Basis    -   11.3 Extensions to the General Replication Theorem    -   11.4 Mathematical Restrictions for the Equilibrium    -   11.5 Examples of DBAR Equilibria with the Digital Replicating        Basis and the Vanilla Replicating Basis    -   Appendix 11A: Proof of General Replication Theorem in Section        11.2.3    -   Appendix 11B: Derivatives of the Self-Hedging Theorem of Section        11.4.5    -   Appendix 11C: Probability Weighted Statistics from Sections        11.5.2 and 11.5.3    -   Appendix 11D: Notation Used in the Body of Text

12 Detailed Description of the Drawings in FIGS. 1 to 28

13 DBAR System Architecture (and Description of the Drawings in FIGS. 32to 68)

-   -   13.1 Terminology and Notation    -   13.2 Overview    -   13.3 Application Architecture    -   13.4 Data    -   13.5 Auction and Event Configuration    -   13.6 Order Processing    -   13.7 Auction State    -   13.8 Startup    -   13.9 CE (calculation engine) implementation    -   13.10 LE (limit order book engine) implementation    -   13.11 Network Architecture    -   13.12 FIGS. 32-68 Legend

14 Appendix 13A: Descriptions of Element Names in DBAR SystemArchitecture

15 Advantages of Preferred Embodiments

-   -   15 Enhanced Parimutuel Wagering    -   15.1 Background and Summary of Example Embodiments    -   15.2 Details and Mathematics of Enhanced Parimutuel Wagering    -   15.3 Horse-Racing Example    -   15.4 Additional Examples of Enhanced Parimutuel Wagering    -   Appendix 15: Notation Used in Section 15

16 Technical Appendix

17 Conclusion

In this specification, including the description of preferred or exampleembodiments of the present invention, specific terminology will be usedfor the sake of clarity. However, the invention is not intended to belimited to the specific terms so used, and it is to be understood thateach specific term includes all equivalents.

1. OVERVIEW: EXCHANGES AND MARKETS FOR DBAR CONTINGENT CLAIMS 1.1Exchange Design

This section describes preferred methods for structuring DBAR contingentclaims and for designing exchanges for the trading of such claims. Thedesign of the exchange is important for effective contingent claimsinvestment in accordance with the present invention. Preferredembodiments of such systems include processes for establishing definedstates and allocating returns, as described below.

-   -   (a) Establishing Defined States and Strikes: In preferred        embodiments, a distribution of possible outcomes for an        observable event is partitioned into defined ranges or states,        and strikes can be established corresponding to measurable        outcomes which occur at one of an upper and/or a lower end of        each defined range or state. In certain preferred embodiments,        one state always occurs because the states are mutually        exclusive and collectively exhaustive. Traders in such an        embodiment invest on their expectation of a return resulting        from the occurrence of a particular outcome within a selected        state. Such investments allow traders to hedge the possible        outcomes of real-world events of economic significance        represented by the states. In preferred embodiments of a group        of DBAR contingent claims, unsuccessful trades or investments        finance the successful trades or investments. In such        embodiments the states for a given contingent claim preferably        are defined in such a way that the states are mutually exclusive        and form the basis of a probability distribution, namely, the        sum of the probabilities of all the uncertain outcomes is unity.        For example, states corresponding to stock price closing values        can be established to support a group of DBAR contingent claims        by partitioning the distribution of possible closing values for        the stock on a given future date into ranges. The distribution        of future stock prices, discretized in this way into defined        states, forms a probability distribution in the sense that each        state is mutually exclusive, and the sum of the probabilities of        the stock closing within each defined state or between two        strikes surrounding the defined state, at the given date is        unity.        -   In preferred embodiments, traders can simultaneously invest            in selected multiple states or strikes within a given            distribution, without immediately breaking up their            investment to fit into each defined states or strikes            selected for investment. Traders thus may place multi-state            or multi-strike investments in order to replicate a desired            distribution of returns from a group of contingent claims.            This may be accomplished in a preferred embodiment of a DBAR            exchange through the use of suspense accounts in which            multi-state or multi-strike investments are tracked and            reallocated periodically as returns adjust in response to            amounts invested during a trading period. At the end of a            given trading period, a multi-state or multi-strike            investment may be reallocated to achieve the desired            distribution of payouts based upon the final invested            amounts across the distribution of states or strikes. Thus,            in such a preferred embodiment, the invested amount            allocated to each of the selected states or strikes, and the            corresponding respective returns, are finalized only at the            closing of the trading period. An example of a multi-state            investment illustrating the use of such a suspense account            is provided in Example 3.1.2, below. Other examples of            multi-state investments are provided in Section 6, below,            which describes embodiments of the present invention that            implement DBAR Digital Options Exchanges. Other examples of            investments in derivatives strategies with multiple strikes            are shown and discussed below, including, inter alia, in            Sections 10 and 11.    -   (b) Allocating Returns: In a preferred embodiment of a group of        DBAR contingent claims according to the present invention,        returns for each state are specified. In such an embodiment,        while the amount invested for a given trade may be fixed, the        return is adjustable. Determination of the returns for a        particular state can be a simple function of the amount invested        in that state and the total amount invested for all of the        defined states for a group of contingent claims. However,        alternate preferred embodiments can also accommodate methods of        return determination that include other factors in addition to        the invested amounts. For example, in a group of DBAR contingent        claims where unsuccessful investments fund returns to successful        investments, the returns can be allocated based on the relative        amounts invested in each state and also on properties of the        outcome, such as the magnitude of the price changes in        underlying securities. An example in section 3.2 below        illustrates such an embodiment in the context of a securities        portfolio.    -   (c) Determining Investment Amounts: In other embodiments, a        group of DBAR contingent claims can be modeled as digital        options, providing a predetermined or defined payout if they        expire in-the-money, and providing no payout if they expire        out-of-the-money. In this embodiment, the investor or trader        specifies a requested payout for a DBAR digital option, and        selects the outcomes for which the digital option will expire        “in the money,” and can specify a limit on the amount they wish        to invest in such a digital option. Since the payout amount per        digital option (or per an order for a digital option) is        predetermined or defined, investment amounts for each digital        option are determined at the end of the trading period along        with the allocation of payouts per digital option as a function        of the requested payouts, selected outcomes (and limits on        investment amounts, if any) for each of the digital options        ordered during the trading period, and the total amount invested        in the auction or market. This embodiment is described in        Section 7 below, along with another embodiment of demand-based        markets or auctions for digital options described in Section 6        below. In additional embodiments, a variety of contingent        claims, including derivatives strategies and financial products        and structured instruments can be replicated or approximated        with a set of DBAR contingent claims (sometimes called,        “replicating claims,”) otherwise regarded as mapping the        contingent claims into a DBAR contingent claim space or basis.        The DBAR contingent claims or replicating claims, can include        replicating digital options or, in a vanilla replicating basis,        include replicating vanilla options alone, or together with        replicating digital options. The price of such replicated        contingent claims is determined by engaging in the demand-based        or DBAR valuation of each of the replicating digital options        and/or vanilla options in the replication set. These embodiments        are described in Sections 10 and 11, as well as a system        architecture described in Section 13 to accomplish a technical        implementation of the entire process.

1.2 Market Operation

-   -   (a) Termination Criteria: In a preferred embodiment of a method        of the present invention, returns to investments in the        plurality of defined states are allocated (and in another        embodiment for DBAR digital options, investment amounts are        determined) after the fulfillment of one or more predetermined        termination criteria. In preferred embodiments, these criteria        include the expiration of a “trading period” and the        determination of the outcome of the relevant event after an        “observation period.” In the trading period, traders invest on        their expectation of a return resulting from the occurrence of a        particular outcome within a selected defined state, such as the        state that IBM stock will close between 120 and 125 on Jul.        6, 1999. In a preferred embodiment, the duration of the trading        period is known to all participants; returns associated with        each state vary during the trading period with changes in        invested amounts; and returns are allocated based on the total        amount invested in all states relative to the amounts invested        in each of the states as at the end of the trading period.        -   Alternatively, the duration of the trading period can be            unknown to the participants. The trading period can end, for            example, at a randomly selected time. Additionally, the            trading period could end depending upon the occurrence of            some event associated or related to the event of economic            significance, or upon the fulfillment of some criterion. For            example, for DBAR contingent claims traded on reinsurance            risk (discussed in Section 3 below), the trading period            could close after an nth catastrophic natural event (e.g., a            fourth hurricane), or after a catastrophic event of a            certain magnitude (e.g., an earthquake of a magnitude of 5.5            or higher on the Richter scale). The trading period could            also close after a certain volume, amount, or frequency of            trading is reached in a respective auction or market.        -   The observation period can be provided as a time period            during which the contingent events are observed and the            relevant outcomes determined for the purpose of allocating            returns. In a preferred embodiment, no trading occurs during            the observation period.        -   The expiration date, or “expiration,” of a group of DBAR            contingent claims as used in this specification occurs when            the termination criteria are fulfilled for that group of            DBAR contingent claims. In a preferred embodiment, the            expiration is the date, on or after the occurrence of the            relevant event, when the outcome is ascertained or observed.            This expiration is similar to well-known expiration features            in traditional options or futures in which a future date,            i.e., the expiration date, is specified as the date upon            which the value of the option or future will be determined            by reference to the value of the underlying financial            product on the expiration date.        -   The duration of a contingent claim as defined for purposes            of this specification is simply the amount of time remaining            until expiration from any given reference date. A trading            start date (“TSD”) and a trading end date (“TED”), as used            in the specification, refer to the beginning and end of a            time period (“trading period”) during which traders can make            investments in a group of DBAR contingent claims. Thus, the            time during which a group of DBAR contingent claims is open            for investment or trading, i.e., the difference between the            TSD and TED, may be referred to as the trading period. In            preferred embodiments, there can be one or many trading            periods for a given expiration date, opening successively            through time. For example, one trading period's TED may            coincide exactly with the subsequent trading period's TSD,            or in other examples, trading periods may overlap.        -   The relationship between the duration of a contingent claim,            the number of trading periods employed for a given event,            and the length and timing of the trading periods, can be            arranged in a variety of ways to maximize trading or achieve            other goals. In preferred embodiments at least one trading            period occurs—that is, starts and ends—prior in time to the            identification of the outcome of the relevant event. In            other words, in preferred embodiments, the trading period            will most likely temporally precede the event defining the            claim. This need not always be so, since the outcome of an            event may not be known for some time thereby enabling            trading periods to end (or even start) subsequent to the            occurrence of the event, but before its outcome is known.        -   A nearly continuous or “quasi-continuous” market can be made            available by creating multiple trading periods for the same            event, each having its own closing returns. Traders can make            investments during successive trading periods as the returns            change. In this way, profits-and-losses can be realized at            least as frequently as in current derivatives markets. This            is how derivatives traders currently are able to hedge            options, futures, and other derivatives trades. In preferred            embodiments of the present invention, traders may be able to            realize profits and at varying frequencies, including more            frequently than daily.    -   (b) Market Efficiency and Fairness: Market prices reflect, among        other things, the distribution of information available to        segments of the participants transacting in the market. In most        markets, some participants will be better informed than others.        In house-banking or traditional markets, market makers protect        themselves from more informed counterparties by increasing their        bid-offer spreads.        -   In preferred embodiments of DBAR contingent claim markets,            there may be no market makers as such who need to protect            themselves. It may nevertheless be necessary to put in place            methods of operation in such markets in order to prevent            manipulation of the outcomes underlying groups of DBAR            contingent claims or the returns payable for various            outcomes. One such mechanism is to introduce an element of            randomness as to the time at which a trading period closes.            Another mechanism to minimize the likelihood and effects of            market manipulation is to introduce an element of randomness            to the duration of the observation period. For example, a            DBAR contingent claim might settle against an average of            market closing prices during a time interval that is            partially randomly determined, as opposed to a market            closing price on a specific day.        -   Additionally, in preferred embodiments incentives can be            employed in order to induce traders to invest earlier in a            trading period rather than later. For example, a DRF may be            used which allocates slightly higher returns to earlier            investments in a successful state than later investments in            that state. For DBAR digital options, an OPF may be used            which determines slightly lower (discounted) prices for            earlier investments than later investments. Earlier            investments may be valuable in preferred embodiments since            they work to enhance liquidity and promote more uniformly            meaningful price information during the trading period.    -   (c) Credit Risk: In preferred embodiments of a DBAR contingent        claims market, the dealer or exchange is substantially protected        from primary market risk by the fundamental principle underlying        the operation of the system—that returns to successful        investments are funded by losses from unsuccessful investments.        The credit risk in such preferred embodiments is distributed        among all the market participants. If, for example, leveraged        investments are permitted within a group of DBAR contingent        claims, it may not be possible to collect the leveraged        unsuccessful investments in order to distribute these amounts        among the successful investments.        -   In almost all such cases there exists, for any given trader            within a group of DBAR contingent claims, a non-zero            possibility of default, or credit risk. Such credit risk is,            of course, ubiquitous to all financial transactions            facilitated with credit.        -   One way to address this risk is to not allow leveraged            investments within the group of DBAR contingent claims,            which is a preferred embodiment of the system and methods of            the present invention. In other preferred embodiments,            traders in a DBAR exchange may be allowed to use limited            leverage, subject to real-time margin monitoring, including            calculation of a trader's impact on the overall level of            credit risk in the DBAR system and the particular group of            contingent claims. These risk management calculations should            be significantly more tractable and transparent than the            types of analyses credit risk managers typically perform in            conventional derivatives markets in order to monitor            counterparty credit risk.        -   An important feature of preferred embodiments of the present            invention is the ability to provide diversification of            credit risk among all the traders who invest in a group of            DBAR contingent claims. In such embodiments, traders make            investments (in the units of value as defined for the group)            in a common distribution of states in the expectation of            receiving a return if a given state is determined to have            occurred. In preferred embodiments, all traders, through            their investments in defined states for a group of            contingent claims, place these invested amounts with a            central exchange or intermediary which, for each trading            period, pays the returns to successful investments from the            losses on unsuccessful investments. In such embodiments, a            given trader has all the other traders in the exchange as            counterparties, effecting a mutualization of counterparties            and counterparty credit risk exposure. Each trader therefore            assumes credit risk to a portfolio of counterparties rather            than to a single counterparty.        -   Preferred embodiments of the DBAR contingent claim and            exchange of the present invention present four principal            advantages in managing the credit risk inherent in leveraged            transactions. First, a preferred form of DBAR contingent            claim entails limited liability investing. Investment            liability is limited in these embodiments in the sense that            the maximum amount a trader can lose is the amount invested.            In this respect, the limited liability feature is similar to            that of a long option position in the traditional markets.            By contrast, a short option position in traditional markets            represents a potentially unlimited liability investment            since the downside exposure can readily exceed the option            premium and is, in theory, unbounded. Importantly, a group            of DBAR contingent claims of the present invention can            easily replicate returns of a traditional short option            position while maintaining limited liability. The limited            liability feature of a group of DBAR contingent claims is a            direct consequence of the demand-side nature of the market.            More specifically, in preferred embodiments there are no            sales or short positions as there are in the traditional            markets, even though traders in a group of DBAR contingent            claims may be able to attain the return profiles of            traditional short positions.        -   Second, in preferred embodiments, a trader within a group of            DBAR contingent claims should have a portfolio of            counterparties as described above. As a consequence, there            should be a statistical diversification of the credit risk            such that the amount of credit risk borne by any one trader            is, on average (and in all but exceptionally rare cases),            less than if there were an exposure to a single counterparty            as is frequently the case in traditional markets. In other            words, in preferred embodiments of the system and methods of            the present invention, each trader is able to take advantage            of the diversification effect that is well known in            portfolio analysis.        -   Third, in preferred embodiments of the present invention,            the entire distribution of margin loans, and the aggregate            amount of leverage and credit risk existing for a group of            DBAR contingent claims, can be readily calculated and            displayed to traders at any time before the fulfillment of            all of the termination criteria for the group of claims.            Thus, traders themselves may have access to important            information regarding credit risk. In traditional markets            such information is not readily available.        -   Fourth, preferred embodiments of a DBAR contingent claim            exchange provide more information about the distribution of            possible outcomes than do traditional market exchanges.            Thus, as a byproduct of DBAR contingent claim trading            according to preferred embodiments, traders have more            information about the distribution of future possible            outcomes for real-world events, which they can use to manage            risk more effectively. For many traders, a significant part            of credit risk is likely to be caused by market risk. Thus,            in preferred embodiments of the present invention, the            ability through an exchange or otherwise to control or at            least provide information about market risk should have            positive feedback effects for the management of credit risk.

A simple example of a group of DBAR contingent claims with the followingassumptions, illustrates some of these features. The example uses thefollowing basic assumptions:

-   -   two defined states (with predetermined termination        criteria): (i) stock price appreciates in one month; (ii) stock        price depreciates in one month; and    -   $100 has been invested in the appreciate state, and $95 in the        depreciate state.

If a trader then invests $1 in the appreciate state, if the stock infact appreciates in the month, then the trader will be allocated apayout of $1.9406 (=196/101)—a return of $0.9406 plus the original $1investment (ignoring, for the purpose of simplicity in thisillustration, a transaction fee). If, before the close of the tradingperiod the trader desires effectively to “sell” his investment in theappreciate state, he has two choices. He could sell the investment to athird party, which would necessitate crossing of a bid and an offer in atwo-way order crossing network. Or, in a preferred embodiment of themethod of the present invention, the trader can invest in the depreciatestate, in proportion to the amount that had been invested in that statenot counting the trader's “new” investments. In this example, in orderto fully hedge his investment in the appreciate state, the trader caninvest $0.95 (95/100) in the depreciate state. Under either possibleoutcome, therefore, the trader will receive a payout of $1.95, i.e., ifthe stock appreciates the trader will receive 196.95/101=$1.95 and ifthe stock depreciates the trader will receive (196.95/95.95)*0.95=$1.95.

1.3 Network Implementation

A market or exchange for groups of DBAR contingent claims marketaccording to the invention is not designed to establish acounterparty-driven or order-matched market. Buyers' bids and sellers'offers do not need to be “crossed.” As a consequence of the absence of aneed for an order crossing network, preferred embodiments of the presentinvention are particularly amenable to large-scale electronic networkimplementation on a wide area network or a private network (with, e.g.,dedicated circuits) or the public Internet, for example. Additionally, anetwork implementation of the embodiments in which contingent claims aremapped or replicated into a vanilla replicating basis, in order to besubject to a demand-based or DBAR valuation, is described in more detailin Section 13 below.

Preferred embodiments of an electronic network-based embodiment of themethod of trading in accordance with the invention include one or moreof the following features.

-   -   (a) User Accounts: DBAR contingent claims investment accounts        are established using electronic methods.    -   (b) Interest and Margin Accounts: Trader accounts are maintained        using electronic methods to record interest paid to traders on        open DBAR contingent claim balances and to debit trader balances        for margin loan interest. Interest is typically paid on        outstanding investment balances for a group of DBAR contingent        claims until the fulfillment of the termination criteria.        Interest is typically charged on outstanding margin loans while        such loans are outstanding. For some contingent claims, trade        balance interest can be imputed into the closing returns of a        trading period.    -   (c) Suspense Accounts: These accounts relate specifically to        investments which have been made by traders, during trading        periods, simultaneously in multiple states for the same event.        Multi-state trades are those in which amounts are invested over        a range of states so that, if any of the states occurs, a return        is allocated to the trader based on the closing return for the        state which in fact occurred. DBAR digital options of the        present invention, described in Section 6, provide other        examples of multi-state trades.        -   A trader can, of course, simply break-up or divide the            multi-state investment into many separate, single-state            investments, although this approach might require the trader            to keep rebalancing his portfolio of single state            investments as returns adjust throughout the trading period            as amounts invested in each state change.        -   Multi-state trades can be used in order to replicate any            arbitrary distribution of payouts that a trader may desire.            For example, a trader might want to invest in all states in            excess of a given value or price for a security underlying a            contingent claim, e.g., the occurrence that a given stock            price exceeds 100 at some future date. The trader might also            want to receive an identical payout no matter what state            occurs among those states. For a group of DBAR contingent            claims there may well be many states for outcomes in which            the stock price exceeds 100 (e.g., greater than 100 and less            than or equal to 101; greater than 101 and less than or            equal to 102, etc.). In order to replicate a multi-state            investment using single state investments, a trader would            need continually to rebalance the portfolio of single-state            investments so that the amount invested in the selected            multi-states is divided among the states in proportion to            the existing amount invested in those states. Suspense            accounts can be employed so that the exchange, rather than            the trader, is responsible for rebalancing the portfolio of            single-state investments so that, at the end of the trading            period, the amount of the multi-state investment is            allocated among the constituent states in such a way so as            to replicate the trader's desired distribution of payouts.            Example 3.1.2 below illustrates the use of suspense accounts            for multi-state investments.    -   (d) Authentication: Each trader may have an account that may be        authenticated using authenticating data.    -   (e) Data Security: The security of contingent claims        transactions over the network may be ensured, using for example        strong forms of public and private key encryption.    -   (f) Real-Time Market Data Server: Real-time market data may be        provided to support frequent calculation of returns and to        ascertain the outcomes during the observation periods.    -   (g) Real-Time Calculation Engine Server: Frequent calculation of        market returns may increase the efficient functioning of the        market. Data on coupons, dividends, market interest rates, spot        prices, and other market data can be used to calculate opening        returns at the beginning of a trading period and to ascertain        observable events during the observation period. Sophisticated        simulation methods may be required for some groups of DBAR        contingent claims in order to estimate expected returns, at        least at the start of a trading period.    -   (h) Real-Time Risk Management Server: In order to compute trader        margin requirements, expected returns for each trader should be        computed frequently. Calculations of “value-at-risk” in        traditional markets can involve onerous matrix calculations and        Monte Carlo simulations. Risk calculations in preferred        embodiments of the present invention are simpler, due to the        existence of information on the expected returns for each state.        Such information is typically unavailable in traditional capital        and reinsurance markets.    -   (i) Market Data Storage: A DBAR contingent claims exchange in        accordance with the invention may generate valuable data as a        byproduct of its operation. These data are not readily available        in traditional capital or insurance markets. In a preferred        embodiment of the present invention, investments may be        solicited over ranges of outcomes for market events, such as the        event that the 30-year U.S. Treasury bond will close on a given        date with a yield between 6.10% and 6.20%. Investment in the        entire distribution of states generates data that reflect the        expectations of traders over the entire distribution of possible        outcomes. The network implementation disclosed in this        specification may be used to capture, store and retrieve these        data.    -   (j) Market Evaluation Server: Preferred embodiments of the        method of the present invention include the ability to improve        the market's efficiency on an ongoing basis. This may readily be        accomplished, for example, by comparing the predicted returns on        a group of DBAR contingent claims returns with actual realized        outcomes. If investors have rational expectations, then DBAR        contingent claim returns will, on average, reflect trader        expectations, and these expectations will themselves be realized        on average. In preferred embodiments, efficiency measurements        are made on defined states and investments over the entire        distribution of possible outcomes, which can then be used for        statistical time series analysis with realized outcomes. The        network implementation of the present invention may therefore        include analytic servers to perform these analyses for the        purpose of continually improving the efficiency of the market.

2. FEATURES OF DBAR CONTINGENT CLAIMS

In a preferred embodiment, a group of a DBAR contingent claims relatedto an observable event includes one or more of the following features:

-   -   (1) A defined set of collectively exhaustive states representing        possible real-world outcomes related to an observable event. In        preferred embodiments, the events are events of economic        significance. The possible outcomes can typically be units of        measurement associated with the event, e.g., an event of        economic interest can be the closing index level of the S&P 500        one month in the future, and the possible outcomes can be entire        range of index levels that are possible in one month. In a        preferred embodiment, the states are defined to correspond to        one or more of the possible outcomes over the entire range of        possible outcomes, so that defined states for an event form a        countable and discrete number of ranges of possible outcomes,        and are collectively exhaustive in the sense of spanning the        entire range of possible outcomes. For example, in a preferred        embodiment, possible outcomes for the S&P 500 can range from        greater than 0 to infinity (theoretically), and a defined state        could be those index values greater than 1000 and less than or        equal to 1100. In such preferred embodiments, exactly one state        occurs when the outcome of the relevant event becomes known.    -   (2) The ability for traders to place trades on the designated        states during one or more trading periods for each event. In a        preferred embodiment, a DBAR contingent claim group defines the        acceptable units of trade or value for the respective claim.        Such units may be dollars, barrels of oil, number of shares of        stock, or any other unit or combination of units accepted by        traders and the exchange for value.    -   (3) An accepted determination of the outcome of the event for        determining which state or states have occurred. In a preferred        embodiment, a group of DBAR contingent claims defines the means        by which the outcome of the relevant events is determined. For        example, the level that the S&P 500 Index actually closed on a        predetermined date would be an outcome observation which would        enable the determination of the occurrence of one of the defined        states. A closing value of 1050 on that date, for instance,        would allow the determination that the state between 1000 and        1100 occurred.    -   (4) The specification of a DRF which takes the traded amount for        each trader for each state across the distribution of states as        that distribution exists at the end of each trading period and        calculates payouts for each investments in each state        conditioned upon the occurrence of each state. In preferred        embodiments, this is done so that the total amount of payouts        does not exceed the total amount invested by all the traders in        all the states. The DRF can be used to show payouts should each        state occur during the trading period, thereby providing to        traders information as to the collective level of interest of        all traders in each state.    -   (5) For DBAR digital options, the specification of an OPF which        takes the requested payout and selection of outcomes and limits        on investment amounts (if any) per digital option at the end of        each trading period and calculates the investment amounts per        digital option, along with the payouts for each digital option        in each state conditioned upon the occurrence of each state. In        this other embodiment, this is done by solving a nonlinear        optimization problem which uses the DRF along with a series of        other parameters to determine an optimal investment amount per        digital option while maximizing the possible payout per digital        option.    -   (6) Payouts to traders for successful investments based on the        total amount of the unsuccessful investments after deduction of        the transaction fee and after fulfillment of the termination        criteria.    -   (7) For DBAR digital options, investment amounts per digital        option after factoring in the transaction fee and after        fulfillment of the termination criteria.

The states corresponding to the range of possible event outcomes arereferred to as the “distribution” or “distribution of states.” Each DBARcontingent claim group or “contract” is typically associated with onedistribution of states. The distribution will typically be defined forevents of economic interest for investment by traders having theexpectation of a return for a reduction of risk (“hedging”), or for anincrease of risk (“speculation”). For example, the distribution can bebased upon the values of stocks, bonds, futures, and foreign exchangerates. It can also be based upon the values of commodity indices,economic statistics (e.g., consumer price inflation monthly reports),property-casualty losses, weather patterns for a certain geographicalregion, and any other measurable or observable occurrence or any otherevent in which traders would not be economically indifferent even in theabsence of a trade on the outcome of the event.

2.1 DBAR Claim Notation

The following notation is used in this specification to facilitatefurther description of DBAR contingent claims:

-   -   m represents the number of traders for a given group of DBAR        contingent claims    -   n represents the number of states for a given distribution        associated with a given group of DBAR contingent claims    -   A represents a matrix with m rows and n columns, where the        element at the i-th row and j-th column, α_(i,j), is the amount        that trader i has invested in state j in the expectation of a        return should state j occur    -   Π represents a matrix with n rows and n columns where element        π_(i,j) is the payout per unit of investment in state i should        state j occur (“unit payouts”)    -   R represents a matrix with n rows and n columns where element        r_(i,j) is the return per unit of investment in state i should        state j occur, i.e., r_(i,j)=π_(i,j)−1 (“unit returns”)    -   P represents a matrix with m rows and n columns, where the        element at the i-th row and j-th column, p_(i,j), is the payout        to be made to trader i should state j occur, i.e., P is equal to        the matrix product A*Π.    -   P*_(j), represents the j-th column of P, for j=1 . . . n, which        contains the payouts to each investment should state j occur    -   P_(i,)* represents the i-th row of P, for i=1 . . . m, which        contains the payouts to trader i    -   s_(i) where i=1 . . . n, represents a state representing a range        of possible outcomes of an observable event.    -   T_(i) where i=1 . . . n, represents the total amount traded in        the expectation of the occurrence of state i    -   T represents the total traded amount over the entire        distribution of states, i.e.,

$T = {\sum\limits_{i = {1\mspace{14mu} \ldots \mspace{14mu} n}}T_{i}}$

-   -   f(A,X) represents the exchange's transaction fee, which can        depend on the entire distribution of traded amounts placed        across all the states as well as other factors, X, some of which        are identified below. For reasons of brevity, for the remainder        of this specification unless otherwise stated, the transaction        fee is assumed to be a fixed percentage of the total amount        traded over all the states.    -   c_(p) represents the interest rate charged on margin loans.    -   c_(r) represents the interest rate paid on trade balances.    -   t represents time from the acceptance of a trade or investment        to the fulfillment of all of the termination criteria for the        group of DBAR contingent claims, typically expressed in years or        fractions thereof    -   X represents other information upon which the DRF or transaction        fee can depend such as information specific to an investment or        a trader, including for example the time or size of a trade.

In preferred embodiments, a DRF is a function that takes the tradedamounts over the distribution of states for a given group of DBARcontingent claims, the transaction fee schedule, and, conditional uponthe occurrence of each state, computes the payouts to each trade orinvestment placed over the distribution of states. In notation, such aDRF is:

P=DRF(A,f(A,X),X|s=s _(i))=AΠ(A,f(A,X),X)  (DRF)

In other words, the m traders who have placed trades across the nstates, as represented in matrix A, will receive payouts as representedin matrix P should state i occur, also, taking into account thetransaction fee f and other factors X. The payouts identified in matrixP can be represented as the product of (a) the payouts per unit tradedfor each state should each state occur, as identified in the matrix Π,and (b) the matrix A which identifies the amounts traded or invested byeach trader in each state. The following notation may be used toindicate that, in preferred embodiments, payouts should not exceed thetotal amounts invested less the transaction fee, irrespective of whichstate occurs:

1_(m) ^(T) *P* _(,j)+ƒ(A,X)<=1_(m) ^(T) *A*1_(n) for j=1 . . . n  (DRFConstraint)

where the 1 represents a column vector with dimension indicated by thesubscript, the superscript T represents the standard transpose operatorand P*_(,j) is the j-th column of the matrix P representing the payoutsto be made to each trader should state j occur. Thus, in preferredembodiments, the unsuccessful investments finance the successfulinvestments. In addition, absent credit-related risks discussed below,in such embodiments there is no risk that payouts will exceed the totalamount invested in the distribution of states, no matter what stateoccurs. In short, a preferred embodiment of a group of DBAR contingentclaims of the present invention is self-financing in the sense that forany state, the payouts plus the transaction fee do not exceed the inputs(i.e., the invested amounts).

The DRF may depend on factors other than the amount of the investmentand the state in which the investment was made. For example, a payoutmay depend upon the magnitude of a change in the observed outcome for anunderlying event between two dates (e.g., the change in price of asecurity between two dates). As another example, the DRF may allocatehigher payouts to traders who initiated investments earlier in thetrading period than traders who invested later in the trading period,thereby providing incentives for liquidity earlier in the tradingperiod. Alternatively, the DRF may allocate higher payouts to largeramounts invested in a given state than to smaller amounts invested forthat state, thereby providing another liquidity incentive.

In any event, there are many possible functional forms for a DRF thatcould be used. To illustrate, one trivial form of a DRF is the case inwhich the traded amounts, A, are not reallocated at all upon theoccurrence of any state, i.e., each trader receives his traded amountback in the event that any state occurs, as indicated by the followingnotation:

P=A if s=s_(i), for i=1 . . . n

This trivial DRF is not useful in allocating and exchanging risk amonghedgers.

For a meaningful risk exchange to occur, a preferred embodiment of a DRFshould effect a meaningful reallocation of amounts invested across thedistribution of states upon the occurrence of at least one state. Groupsof DBAR contingent claims of the present invention are discussed in thecontext of a canonical DRF, which is a preferred embodiment in which theamounts invested in states which did not occur are completelyreallocated to the state which did occur (less any transaction fee). Thepresent invention is not limited to a canonical DRF, and many othertypes of DRFs can be used and may be preferred to implement a group ofDBAR contingent claims. For example, another DRF preferred embodimentallocates half the total amount invested to the outcome state andrebates the remainder of the total amount invested to the states whichdid not occur. In another preferred embodiment, a DRF would allocatesome percentage to an occurring state, and some other percentage to oneor more “nearby” or “adjacent” states with the bulk of the non-occurringstates receiving zero payouts. Section 7 describes an OPF for DBARdigital options which includes a DRF and determines investment amountsper investment or order along with allocating returns. Other DRFs willbe apparent to those of skill in the art from review of thisspecification and practice of the present invention.

2.2 Units of Investments and Payouts

The units of investments and payouts in systems and methods of thepresent invention may be units of currency, quantities of commodities,numbers of shares of common stock, amount of a swap transaction or anyother units representing economic value. Thus, there is no limitationthat the investments or payouts be in units of currency or money (e.g.,U.S. dollars) or that the payouts resulting from the DRF be in the sameunits as the investments. Preferably, the same unit of value is used torepresent the value of each investment, the total amount of allinvestments in a group of DBAR contingent claims, and the amountsinvested in each state.

It is possible, for example, for traders to make investments in a groupof DBAR contingent claims in numbers of shares of common stock and forthe applicable DRF (or OPF) to allocate payouts to traders in JapaneseYen or barrels of oil. Furthermore, it is possible for traded amountsand payouts to be some combination of units, such as, for example, acombination of commodities, currencies, and number of shares. Inpreferred embodiments traders need not physically deposit or receivedelivery of the value units, and can rely upon the DBAR contingent claimexchange to convert between units for the purposes of facilitatingefficient trading and payout transactions. For example, a DBARcontingent claim might be defined in such a way so that investments andpayouts are to be made in ounces of gold. A trader can still depositcurrency, e.g., U.S. dollars, with the exchange and the exchange can beresponsible for converting the amount invested in dollars into thecorrect units, e.g., gold, for the purposes of investing in a givenstate or receiving a payout. In this specification, a U.S. dollar istypically used as the unit of value for investments and payouts. Thisinvention is not limited to investments or payouts in that value unit.In situations where investments and payouts are made in different unitsor combinations of units, for purpose of allocating returns to eachinvestment the exchange preferably converts the amount of eachinvestment, and thus the total of the investments in a group of DBARcontingent claims, into a single unit of value (e.g., dollars). Example3.1.20 below illustrates a group of DBAR contingent claims in whichinvestments and payouts are in units of quantities of common stockshares.

2.3 Canonical Demand Reallocation Function

A preferred embodiment of a DRF that can be used to implement a group ofDBAR contingent claims is termed a “canonical” DRF. A canonical DRF is atype of DRF which has the following property: upon the occurrence of agiven state i, investors who have invested in that state receive apayout per unit invested equal to (a) the total amount traded for allthe states less the transaction fee, divided by (b) the total amountinvested in the occurring state. A canonical DRF may employ atransaction fee which may be a fixed percentage of the total amounttraded, T, although other transaction fees are possible. Traders whomade investments in states which not did occur receive zero payout.Using the notation developed above:

$\pi_{i,j} = \frac{\left( {1 - f} \right)*T}{T_{i}}$

if i=j, i.e., the unit payout to an investment in state i if state ioccurs π_(i,j)=0 otherwise, i.e., if i≠j, so that the payout is zero toinvestments in state i if state j occurs.In a preferred embodiment of a canonical DRF, the unit payout matrix Πas defined above is therefore a diagonal matrix with entries equal toπ_(i,j) for i=j along the diagonal, and zeroes for all off-diagonalentries. For example, in a preferred embodiment for n=5 states, the unitpayout matrix is:

$\begin{matrix}{\prod{= {\begin{bmatrix}\frac{T}{T_{1}} & 0 & 0 & 0 & 0 \\0 & \frac{T}{T_{2}} & 0 & 0 & 0 \\0 & 0 & \frac{T}{T_{3}} & 0 & 0 \\0 & 0 & 0 & \frac{T}{T_{4}} & 0 \\0 & 0 & 0 & 0 & \frac{T}{T_{5}}\end{bmatrix}*\left( {1 - f} \right)}}} \\{= {\begin{bmatrix}\frac{1}{T_{1}} & 0 & 0 & 0 & 0 \\0 & \frac{1}{T_{2}} & 0 & 0 & 0 \\0 & 0 & \frac{1}{T_{3}} & 0 & 0 \\0 & 0 & 0 & \frac{1}{T_{4}} & 0 \\0 & 0 & 0 & 0 & \frac{1}{T_{5}}\end{bmatrix}*T*\left( {1 - f} \right)}}\end{matrix}$

For this embodiment of a canonical DRF, the payout matrix is the totalamount invested less the transaction fee, multiplied by a diagonalmatrix which contains the inverse of the total amount invested in eachstate along the diagonal, respectively, and zeroes elsewhere. Both T,the total amount invested by all m traders across all n states, andT_(i), the total amount invested in state i, are functions of the matrixA, which contains the amount each trader has invested in each state:

T _(i)=1_(m) ^(T) *A*B _(n)(i)

T=1_(m) ^(T) *A*1_(n)

where B_(n)(i) is a column vector of dimension n which has a 1 at thei-th row and zeroes elsewhere. Thus, with n=5 as an example, thecanonical DRF described above has a unit payout matrix which is afunction of the amounts traded across the states and the transactionfee:

$\prod{= {\quad {\begin{bmatrix}\frac{1}{1_{m}^{T}*A*{B_{n}(1)}} & 0 & 0 & 0 & 0 \\0 & \frac{1}{1_{m}^{T}*A*{B_{n}(2)}} & 0 & 0 & 0 \\0 & 0 & \frac{1}{1_{m}^{T}*A*{B_{n}(3)}} & 0 & 0 \\0 & 0 & 0 & \frac{1}{1_{m}^{T}*A*{B_{n}(4)}} & 0 \\0 & 0 & 0 & 0 & \frac{1}{1_{m}^{T}*A*{B_{n}(5)}}\end{bmatrix}*1_{m}^{T}*A*1_{n}*\left( {1 - f} \right)}}}$

which can be generalized for any arbitrary number of states. The actualpayout matrix, in the defined units of value for the group of DBARcontingent claims (e.g., dollars), is the product of the m×n tradedamount matrix A and the n×n unit payout matrix Π, as defined above:

P=A*Π(A,ƒ)  (CDRF)

This provides that the payout matrix as defined above is the matrixproduct of the amounts traded as contained in the matrix A and the unitpayout matrix Π, which is itself a function of the matrix A and thetransaction fee, f. The expression is labeled CDRF for “Canonical DemandReallocation Function.”

It should be noted that, in this preferred embodiment, any change to thematrix A will generally have an effect on any given trader's payout,both due to changes in the amount invested, i.e., a direct effectthrough the matrix A in the CDRF, and changes in the unit payouts, i.e.,an indirect effect since the unit payout matrix Π is itself a functionof the traded amount matrix A.

2.4 Computing Investment Amounts to Achieve Desired Payouts

In preferred embodiments of a group of DBAR contingent claims of thepresent invention, some traders make investments in states during thetrading period in the expectation of a payout upon the occurrence of agiven state, as expressed in the CDRF above. Alternatively, a trader mayhave a preference for a desired payout distribution should a given stateoccur. DBAR digital options, described in Section 6, are an example ofan investment with a desired payout distribution should one or morespecified states occur. Such a payout distribution could be denotedP_(i,*), which is a row corresponding to trader i in payout matrix P.Such a trader may want to know how much to invest in contingent claimscorresponding to a given state or states in order to achieve this payoutdistribution. In a preferred embodiment, the amount or amounts to beinvested across the distribution of states for the CDRF, given a payoutdistribution, can be obtained by inverting the expression for the CDRFand solving for the traded amount matrix A:

A=P*Π(A,ƒ)⁻¹  (CDRF 2)

In this notation, the −1 superscript on the unit payout matrix denotes amatrix inverse.

Expression CDRF 2 does not provide an explicit solution for the tradedamount matrix A, since the unit payout matrix Π is itself a function ofthe traded amount matrix. CDRF 2 typically involves the use of numericalmethods to solve m simultaneous quadratic equations. For example,consider a trader who would like to know what amount, α, should betraded for a given state i in order to achieve a desired payout of p.Using the “forward” expression to compute payouts from traded amounts asin CDRF above yields the following equation:

$p = {\left( \frac{T + \alpha}{T_{i} + \alpha} \right)*\alpha}$

This represents a given row and column of the matrix equation CDRF afterα has been traded for state i (assuming no transaction fee). Thisexpression is quadratic in the traded amount α, and can be solved forthe positive quadratic root as follows:

$\begin{matrix}{\alpha = \frac{\left( {p - T} \right) + \sqrt{\left( {p - T} \right)^{2} + {4*p*T_{i}}}}{2}} & \left( {{CDRF}\mspace{14mu} 3} \right)\end{matrix}$

2.5 A Canonical DRF Example

A simplified example illustrates the use of the CDRF with a group ofDBAR contingent claims defined over two states (e.g., states “1” and“2”) in which four traders make investments. For the example, thefollowing assumptions are made: (1) the transaction fee, f, is zero; (2)the investment and payout units are both dollars; (3) trader 1 has madeinvestments in the amount of $5 in state 1 and $10 state 2; and (4)trader 2 has made an investment in the amount of $7 for state 1 only.With the investment activity so far described, the traded amount matrixA, which as 4 rows and 2 columns, and the unit payout matrix Π which has2 rows and 2 columns, would be denoted as follows:

${A = \begin{matrix}5 & 10 \\7 & 0 \\0 & 0 \\0 & 0\end{matrix}}\mspace{14mu}$ $\Pi = {\begin{bmatrix}\frac{1}{12} & 0 \\0 & \frac{1}{10}\end{bmatrix}*22}$

The payout matrix P, which contains the payouts in dollars for eachtrader should each state occur is, the product of A and Π:

$P = \begin{matrix}9.167 & 22 \\12.833 & 0 \\0 & 0 \\0 & 0\end{matrix}$

The first row of P corresponds to payouts to trader 1 based on hisinvestments and the unit payout matrix. Should state 1 occur, trader 1will receive a payout of $9.167 and will receive $22 should state 2occur. Similarly, trader 2 will receive $12.833 should state 1 occur and$0 should state 2 occur (since trader 2 did not make any investment instate 2). In this illustration, traders 3 and 4 have $0 payouts sincethey have made no investments.

In accordance with the expression above labeled “DRF Constraint,” thetotal payouts to be made upon the occurrence of either state is lessthan or equal to the total amounts invested. In other words, the CDRF inthis example is self-financing so that total payouts plus thetransaction fee (assumed to be zero in this example) do not exceed thetotal amounts invested, irrespective of which state occurs. This isindicated by the following notation:

1_(m) ^(T) *P* _(,1)=22≦1_(m) ^(T) *A*1_(n)=22

1_(m) ^(T) *P* _(,2)=22≦1_(m) ^(T) *A*1_(n)=22

Continuing with this example, it is now assumed that traders 3 and 4each would like to make investments that generate a desired payoutdistribution. For example, it is assumed that trader 3 would like toreceive a payout of $2 should state 1 occur and $4 should state 2 occur,while trader 4 would like to receive a payout of $5 should state 1 occurand $0 should state 2 occur. In the CDRF notation:

P _(3,*)=[2 4]

P _(4,*)=[5 0]

In a preferred embodiment and this example, payouts are made based uponthe invested amounts A, and therefore are also based on the unit payoutmatrix Π(A,f(A)), given the distribution of traded amounts as they existat the end of the trading period. For purposes of this example, it isnow further assumed (a) that at the end of the trading period traders 1and 2 have made investments as indicated above, and (b) that the desiredpayout distributions for traders 3 and 4 have been recorded in asuspense account which is used to determine the allocation ofmulti-state investments to each state in order to achieve the desiredpayout distributions for each trader, given the investments by the othertraders as they exist at the end of the trading period. In order todetermine the proper allocation, the suspense account can be used tosolve CDRF 2, for example:

$\left\lbrack \begin{matrix}5 & 10 \\7 & 0 \\\alpha_{3,1} & \alpha_{3,2} \\\alpha_{4,1} & \alpha_{4,2}\end{matrix} \right\rbrack = {\left\lbrack \begin{matrix}p_{1,1} & p_{1,2} \\p_{2,1} & p_{2,2} \\2 & 4 \\5 & 0\end{matrix} \right\rbrack*\left\lbrack \begin{matrix}\frac{1}{\left( {5 + 7 + \alpha_{3,1} + \alpha_{4,1}} \right)} & 0 \\0 & \frac{1}{\left( {10 + 0 + \alpha_{3,2} + \alpha_{4,2}} \right)}\end{matrix} \right\rbrack \frac{{cont}^{\prime}d}{below}*\left( {5 + 10 + 7 + 0 + \alpha_{3,1} + \alpha_{4,1} + \alpha_{3,2} + \alpha_{4,2}} \right)}$

The solution of this expression will yield the amounts that traders 3and 4 need to invest in for contingent claims corresponding to states 1and 2 to in order to achieve their desired payout distributions,respectively. This solution will also finalize the total investmentamount so that traders 1 and 2 will be able to determine their payoutsshould either state occur. This solution can be achieved using acomputer program that computes an investment amount for each state foreach trader in order to generate the desired payout for that trader forthat state. In a preferred embodiment, the computer program repeats theprocess iteratively until the calculated investment amounts converge,i.e., so that the amounts to be invested by traders 3 and 4 no longermaterially change with each successive iteration of the computationalprocess. This method is known in the art as fixed point iteration and isexplained in more detail in the Technical Appendix. The following tablecontains a computer code listing of two functions written in Microsoft'sVisual Basic which can be used to perform the iterative calculations tocompute the final allocations of the invested amounts in this example ofa group of DBAR contingent claims with a Canonical Demand ReallocationFunction:

TABLE 1 Illustrative Visual Basic Computer Code for Solving CDRF 2Function allocatetrades(A_mat, P_mat) As Variant Dim A_final Dim tradesAs Long Dim states As Long trades = P_mat.Rows.Count states =P_mat.Columns.Count ReDim A_final(1 To trades, 1 To states) ReDimstatedem(1 To states) Dim i As Long Dim totaldemand As Double Dim totaldesired As Double Dim iterations As Long iterations = 10 For i = 1 Totrades For j = 1 To states statedem(j) = A_mat(i, j) + statedem(j)A_final(i, j) = A_mat(i, j) Next j Next i For i = 1 To statestotaldemand = totaldemand + statedem(i) Next i For i = 1 To iterationsFor j = 1 To trades For z = 1 To states If A_mat(j, z) = 0 Thentotaldemand = totaldemand − A_final(j, z) statedem(z) = statedem(z) −A_final(j, z) tempalloc = A_final(j, z) A_final(j, z) =stateall(totaldemand, P_mat(j, z), statedem(z)) totaldemand = A_final(j,z) + totaldemand statedem(z) = A_final(j, z) + statedem(z) End If Next zNext j Next i allocatetrades = A_final End Function Functionstateall(totdemex, despaystate, totstateex) Dim sol1 As Double Sol1 =(−(totdemex − despaystate) + ((totdemex − despaystate) {circumflex over( )} 2 + 4 * despaystate * totstateex) {circumflex over ( )} 0.5) / 2stateall = sol1 End FunctionFor this example involving two states and four traders, use of thecomputer code represented in Table 1 produces an investment amountmatrix A, as follows:

$A = \begin{matrix}5 & 10 \\7 & 0 \\1.1574 & 1.6852 \\2.8935 & 0\end{matrix}$

The matrix of unit payouts, Π, can be computed from A as described aboveand is equal to:

$\Pi = \begin{matrix}1.728 & 0 \\0 & 2.3736\end{matrix}$

The resulting payout matrix P is the product of A and Π and is equal to:

$P = \begin{matrix}8.64 & 23.7361 \\12.0961 & 0 \\2 & 4 \\5 & 0\end{matrix}$

It can be noted that the sum of each column of P, above is equal to27.7361, which is equal (in dollars) to the total amount invested so, asdesired in this example, the group of DBAR contingent claims isself-financing. The allocation is said to be in equilibrium, since theamounts invested by traders 1 and 2 are undisturbed, and traders 3 and 4receive their desired payouts, as specified above, should each stateoccur.

2.6 Interest Considerations

When investing in a group of DBAR contingent claims, traders willtypically have outstanding balances invested for periods of time and mayalso have outstanding loans or margin balances from the exchange forperiods of time. Traders will typically be paid interest on outstandinginvestment balances and typically will pay interest on outstandingmargin loans. In preferred embodiments, the effect of trade balanceinterest and margin loan interest can be made explicit in the payouts,although in alternate preferred embodiments these items can be handledoutside of the payout structure, for example, by debiting and creditinguser accounts. So, if a fraction β of a trade of one value unit is madewith cash and the rest on margin, the unit payout π_(i) in the eventthat state i occurs can be expressed as follows:

$\pi_{i} = {\frac{\left( {1 - f} \right)*T}{T_{i}} + {\beta*\left( c_{r} \right)*t_{b}} - {\left( {1 - \beta} \right)*\left( c_{p} \right)*t_{l}}}$

where the last two terms express the respective credit for tradebalances per unit invested for time t_(b) and debit for margin loans perunit invested for time t₁.

2.7 Returns and Probabilities

In a preferred embodiment of a group of DBAR contingent claims with acanonical DRF, returns which represent the percentage return per unit ofinvestment are closely related to payouts. Such returns are also closelyrelated to the notion of a financial return familiar to investors. Forexample, if an investor has purchased a stock for $100 and sells it for$110, then this investor has realized a return of 10% (and a payout of$110).

In a preferred embodiment of a group of DBAR contingent claims with acanonical DRF, the unit return, r_(i), should state i occur may beexpressed as follows:

$r_{i} = \frac{{\left( {1 - f} \right)*{\sum\limits_{i = {1\mspace{14mu} \ldots \mspace{14mu} n}}T_{i}}} - T_{i}}{T_{i}}$

if state i occurs

-   -   r_(i)=−1 otherwise, i.e., if state i does not occur

In such an embodiment, the return per unit investment in a state thatoccurs is a function of the amount invested in that state, the amountinvested in all the other states and the exchange fee. The unit returnis −100% for a state that does not occur, i.e., the entire amountinvested in the expectation of receiving a return if a state occurs isforfeited if that state fails to occur. A −100% return in such an eventhas the same return profile as, for example, a traditional optionexpiring “out of the money.” When a traditional option expires out ofthe money, the premium decays to zero, and the entire amount invested inthe option is lost.

For purposes of this specification, a payout is defined as one plus thereturn per unit invested in a given state multiplied by the amount thathas been invested in that state. The sum of all payouts P_(s), for agroup of DBAR contingent claims corresponding to all n possible statescan be expressed as follows:

$P_{S} = {{\left( {1 + r_{i}} \right)*T_{i}} + {\sum\limits_{j,{j \neq i}}{\left( {1 + r_{j}} \right)*T_{j}}}}$i, j = 1  …  n

In a preferred embodiment employing a canonical DRF, the payout P_(S)may be found for the occurrence of state i by substituting the aboveexpressions for the unit return in any state:

$\begin{matrix}{P_{S} = {{\left( {\frac{{\left( {1 - f} \right){\sum\limits_{i = {1\mspace{14mu} \ldots \mspace{14mu} n}}T_{i}}} - T_{i}}{T_{i}} + 1} \right)*T_{i}} + {\sum\limits_{j,{j \neq i}}{\left( {{- 1} + 1} \right)*T_{j}}}}} \\{= {\left( {1 - f} \right)*{\sum\limits_{i = {1\mspace{14mu} \ldots \mspace{11mu} n}}T_{i}}}}\end{matrix}$

Accordingly, in such a preferred embodiment, for the occurrence of anygiven state, no matter what state, the aggregate payout to all of thetraders as a whole is one minus the transaction fee paid to the exchange(expressed in this preferred embodiment as a percentage of totalinvestment across all the states), multiplied by the total amountinvested across all the states for the group of DBAR contingent claims.This means that in a preferred embodiment of a group of the DBARcontingent claims, and assuming no credit or similar risks to theexchange, (1) the exchange has zero probability of loss in any givenstate; (2) for the occurrence of any given state, the exchange receivesan exchange fee and is not exposed to any risk; (3) payouts and returnsare a function of demand flow, i.e., amounts invested; and (4)transaction fees or exchange fees can be a simple function of aggregateamount invested.

Other transaction fees can be implemented. For example, the transactionfee might have a fixed component for some level of aggregate amountinvested and then have either a sliding or fixed percentage applied tothe amount of the investment in excess of this level. Other methods fordetermining the transaction fee are apparent to those of skill in theart, from this specification or based on practice of the presentinvention.

In a preferred embodiment, the total distribution of amounts invested inthe various states also implies an assessment by all traderscollectively of the probabilities of occurrence of each state. In apreferred embodiment of a group of DBAR contingent claims with acanonical DRF, the expected return E(r_(i)) for an investment in a givenstate i (as opposed to the return actually received once outcomes areknown) may be expressed as the probability weighted sum of the returns:

E(r _(i))=q _(i) *r _(i)+(1−q _(i))*−1=q _(i)*(1+r _(i))−1

Where q_(i) is the probability of the occurrence of state i implied bythe matrix A (which contains all of the invested amounts for all statesin the group of DBAR contingent claims). Substituting the expression forr_(i) from above yields:

${E\left( r_{i} \right)} = {{q_{i}*\left( \frac{\left( {1 - f} \right)*{\sum\limits_{i}T_{i}}}{T_{i}} \right)} - 1}$

In an efficient market, the expected return E(r_(i)) across all statesis equal to the transaction costs of trading, i.e., on average, alltraders collectively earn returns that do not exceed the costs oftrading. Thus, in an efficient market for a group of DBAR contingentclaims using a canonical, where E(r_(i)) equals the transaction fee, −f,the probability of the occurrence of state i implied by matrix A iscomputed to be:

$q_{i} = \frac{T_{i}}{\sum\limits_{i}T_{i}}$

Thus, in such a group of DBAR contingent claims, the implied probabilityof a given state is the ratio of the amount invested in that statedivided by the total amount invested in all states. This relationshipallows traders in the group of DBAR contingent claims (with a canonicalDRF) readily to calculate the implied probability which traders attachto the various states.

Information of interest to a trader typically includes the amountsinvested per state, the unit return per state, and implied stateprobabilities. An advantage of the DBAR exchange of the presentinvention is the relationship among these quantities. In a preferredembodiment, if the trader knows one, the other two can be readilydetermined. For example, the relationship of unit returns to theoccurrence of a state and the probability of the occurrence of thatstate implied by A can be expressed as follows:

$q_{i} = \frac{\left( {1 - f} \right)}{\left( {1 + r_{i}} \right)}$

The expressions derived above show that returns and implied stateprobabilities may be calculated from the distribution of the investedamounts, T_(i), for all states, i=1 . . . n. In the traditional markets,the amount traded across the distribution of states (limit order book),is not readily available. Furthermore, in traditional markets there areno such ready mathematical calculations that relate with any precisioninvested amounts or the limit order book to returns or prices whichclear the market, i.e., prices at which the supply equals the demand.Rather, in the traditional markets, specialist brokers and market makerstypically have privileged access to the distribution of bids and offers,or the limit order book, and frequently use this privileged informationin order to set market prices that balance supply and demand at anygiven time in the judgment of the market maker.

2.8 Computations when Invested Amounts are Large

In a preferred embodiment of a group of DBAR contingent claims using acanonical DRF, when large amounts are invested across the distributionof states, it may be possible to perform approximate investmentallocation calculations in order to generate desired payoutdistributions. The payout, p, should state i occur for a trader whoconsiders making an investment of size α in state i has been shown aboveto be:

$p = {\left( \frac{T + \alpha}{T_{i} + \alpha} \right)*\alpha}$

If α is small compared to both the total invested in state i and thetotal amount invested in all the states, then adding a to state i willnot have a material effect on the ratio of the total amount invested inall the states to the total amount invested in state i. In thesecircumstances,

$\frac{T + \alpha}{T_{i} + \alpha} \approx \frac{T}{T_{i}}$

Thus, in preferred embodiments where an approximation is acceptable, thepayout to state i may be expressed as:

$p \approx {\frac{T}{T_{i}}*\alpha}$

In these circumstances, the investment needed to generate the payout pis:

${\alpha \approx {\frac{T_{i}}{T}*p}} = {q_{i}*p}$

These expressions indicate that in preferred embodiments, the amount tobe invested to generate a desired payout is approximately equal to theratio of the total amount invested in state i to the total amountinvested in all states, multiplied by the desired payout. This isequivalent to the implied probability multiplied by the desired payout.Applying this approximation to the expression CDRF 2, above, yields thefollowing:

A≈P*Π ⁻¹ =P*Q

where the matrix Q, of dimension n×n, is equal to the inverse of unitpayouts Π and has along the diagonals q_(i) for i=1 . . . n. Thisexpression provides an approximate but more readily calculable solutionto CDRF 2 as the expression implicitly assumes that an amount investedby a trader has approximately no effect on the existing unit payouts orimplied probabilities. This approximate solution, which is linear andnot quadratic, will sometimes be used in the following examples where itcan be assumed that the total amounts invested are large in relation toany given trader's particular investment.

3. EXAMPLES OF GROUPS OF DBAR CONTINGENT CLAIMS 3.1 DBAR RangeDerivatives

A DBAR Range Derivative (DBAR RD) is a type of group of DBAR contingentclaims implemented using a canonical DRF described above (although aDBAR range derivative can also be implemented, for example, for a groupof DBAR contingent claims, including DBAR digital options, based on thesame ranges and economic events established below using, e.g., anon-canonical DRF and an OPF). In a DBAR RD, a range of possibleoutcomes associated with an observable event of economic significance ispartitioned into defined states. In a preferred embodiment, the statesare defined as discrete ranges of possible outcomes so that the entiredistribution of states covers all the possible outcomes—that is, thestates are collectively exhaustive. Furthermore, in a DBAR RD, statesare preferably defined so as to be mutually exclusive as well, meaningthat the states are defined in such a way so that exactly one stateoccurs. If the states are defined to be both mutually exclusive andcollectively exhaustive, the states form the basis of a probabilitydistribution defined over discrete outcome ranges. Defining the statesin this way has many advantages as described below, including theadvantage that the amount which traders invest across the states can bereadily converted into implied probabilities representing the collectiveassessment of traders as to the likelihood of the occurrence of eachstate.

The system and methods of the present invention may also be applied todetermine projected DBAR RD returns for various states at the beginningof a trading period. Such a determination can be, but need not be, madeby an exchange. In preferred embodiments of a group of DBAR contingentclaims the distribution of invested amounts at the end of a tradingperiod determines the returns for each state, and the amount invested ineach state is a function of trader preferences and probabilityassessments of each state. Accordingly, some assumptions typically needto be made in order to determine preliminary or projected returns foreach state at the beginning of a trading period.

An illustration is provided to explain further the operation of DBARRDs. In the following illustration, it is assumed that all traders arerisk neutral so that implied probabilities for a state are equal to theactual probabilities, and so that all traders have identical probabilityassessments of the possible outcomes for the event defining thecontingent claim. For convenience in this illustration, the eventforming the basis for the contingent claims is taken to be a closingprice of a security, such as a common stock, at some future date; andthe states, which represent the possible outcomes of the level of theclosing price, are defined to be distinct, mutually exclusive andcollectively exhaustive of the range of (possible) closing prices forthe security. In this illustration, the following notation is used:

-   -   τ represents a given time during the trading period at which        traders are making investment decisions    -   θ represents the time corresponding to the expiration of the        contingent claim    -   V_(τ) represents the price of underlying security at time τ    -   V_(θ) represents the price of underlying security at time θ    -   Z(τ,θ) represents the present value of one unit of value payable        at time θ evaluated at time τ    -   D(τ,θ) represents dividends or coupons payable between time τ        and θ    -   σ_(t) represents annualized volatility of natural logarithm        returns of the underlying security    -   dz represents the standard normal variate        Traders make choices at a representative time, τ, during a        trading period which is open, so that time τ is temporally        subsequent to the current trading period's TSD.

In this illustration, and in preferred embodiments, the defined statesfor the group of contingent claims for the final closing price V_(θ) areconstructed by discretizing the full range of possible prices intopossible mutually exclusive and collectively exhaustive states. Thetechnique is similar to forming a histogram for discrete countable data.The endpoints of each state can be chosen, for example, to be equallyspaced, or of varying spacing to reflect the reduced likelihood ofextreme outcomes compared to outcomes near the mean or median of thedistribution. States may also be defined in other manners apparent toone of skill in the art. The lower endpoint of a state can be includedand the upper endpoint excluded, or vice versa. In any event, inpreferred embodiments, the states are defined (as explained below) tomaximize the attractiveness of investment in the group of DBARcontingent claims, since it is the invested amounts that ultimatelydetermine the returns that are associated with each defined state.

The procedure of defining states, for example for a stock price, can beaccomplished by assuming lognormality, by using statistical estimationtechniques based on historical time series data and cross-section marketdata from options prices, by using other statistical distributions, oraccording to other procedures known to one of skill in the art orlearned from this specification or through practice of the presentinvention. For example, it is quite common among derivatives traders toestimate volatility parameters for the purpose of pricing options byusing the econometric techniques such as GARCH. Using these parametersand the known dividend or coupons over the time period from τ to θ, forexample, the states for a DBAR RD can be defined. A lognormaldistribution is chosen for this illustration since it is commonlyemployed by derivatives traders as a distributional assumption for thepurpose of evaluating the prices of options and other derivativesecurities. Accordingly, for purposes of this illustration it is assumedthat all traders agree that the underlying distribution of states forthe security are lognormally distributed such that:

${\overset{\sim}{V}}_{\theta} = {\left( {\frac{V_{\tau}}{Z\left( {\tau,\theta} \right)} - \frac{D\left( {\tau,\theta} \right)}{Z\left( {\tau,\theta} \right)}} \right)*^{{{- \sigma^{2}}/2}*{({\theta - \tau})}}*^{\sigma*\sqrt{\theta - \tau}*{dz}}}$

where the “tilde” on the left-hand side of the expression indicates thatthe final closing price of the value of the security at time θ is yet tobe known. Inversion of the expression for dz and discretization ofranges yields the following expressions:

${dz} = {\ln \frac{\left( \frac{V_{\theta}*^{\frac{\sigma^{2}}{2}*{({\theta - \tau})}}}{\left( {\frac{V_{\tau}}{Z\left( {\tau,\theta} \right)} - \frac{D\left( {\tau,\theta} \right)}{Z\left( {\tau,\theta} \right)}} \right)} \right)}{\left( {\sigma*\sqrt{\theta - \tau}} \right)}}$q_(i)(V_(i) <  = V_(θ) < V_(i + 1)) = cdf(dz_(i + 1)) − cdf(dz_(i))${r_{i}\left( {V_{i}<=V_{\theta} < V_{i + 1}} \right)} = {\frac{\left( {1 - f} \right)}{q_{i}\left( {V_{i}<=V_{\theta} < V_{i + 1}} \right)} - 1}$

where cdf(dz) is the cumulative standard normal function.

The assumptions and calculations reflected in the expressions presentedabove can also be used to calculate indicative returns (“openingreturns”), r_(i) at a beginning of the trading period for a given groupof DBAR contingent claims. In a preferred embodiment, the calculatedopening returns are based on the exchange's best estimate of theprobabilities for the states defining the claim and therefore mayprovide good indications to traders of likely returns once trading isunderway. In another preferred embodiment, described with respect toDBAR digital options in Section 6 and another embodiment described inSection 7, a very small number of value units may be used in each stateto initialize the contract or group of contingent claims. Of course,opening returns need not be provided at all, as traded amounts placedthroughout the trading period allows the calculation of actual expectedreturns at any time during the trading period.

The following examples of DBAR range derivatives and other contingentclaims serve to illustrate their operation, their usefulness inconnection with a variety of events of economic significance involvinginherent risk or uncertainty, the advantages of exchanges for groups ofDBAR contingent claims, and, more generally, systems and methods of thepresent invention. Sections 6 and 7 also provide examples of DBARcontingent claims of the present invention that provide profit and lossscenarios comparable to those provided by digital options inconventional options markets, and that can be based on any of thevariety of events of economic significance described in the followingexamples of DBAR RDs.

In each of the examples in this Section, a state is defined to include arange of possible outcomes of an event of economic significance. Theevent of economic significance for any DBAR auction or market (includingany market or auction for DBAR digital options) can be, for example, anunderlying economic event (e.g., price of stock) or a measured parameterrelated to the underlying economic event (e.g., a measured volatility ofthe price of stock). A curved brace “(“or”)” denotes strict inequality(e.g., “greater than” or “less than,” respectively) and a square brace“]” or “[” shall denote weak inequality (e.g., “less than or equal to”or “greater than or equal to,” respectively). For simplicity, and unlessotherwise stated, the following examples also assume that the exchangetransaction fee, f, is zero.

Example 3.1.1 DBAR Contingent Claim On Underlying Common Stock

-   -   Underlying Security: Microsoft Corporation Common Stock (“MSFT”)    -   Date: Aug. 18, 1999    -   Spot Price: 85    -   Market Volatility: 50% annualized    -   Trading Start Date: Aug. 18, 1999, Market Open    -   Trading End Date: Aug. 18, 1999, Market Close    -   Expiration: Aug. 19, 1999, Market Close    -   Event: MSFT Closing Price at Expiration    -   Trading Time: 1 day    -   Duration to TED: 1 day    -   Dividends Payable to Expiration: 0    -   Interbank short-term interest rate to Expiration: 5.5%        (Actual/360 daycount)    -   Present Value factor to Expiration: 0.999847    -   Investment and Payout Units: U.S. Dollars (“USD”)

In this Example 3.1.1, the predetermined termination criteria are theinvestment in a contingent claim during the trading period and theclosing of the market for Microsoft common stock on Aug. 19, 1999.

If all traders agree that the underlying distribution of closing pricesis lognormally distributed with volatility of 50%, then an illustrative“snapshot” distribution of invested amounts and returns for $100 millionof aggregate investment can be readily calculated to yield the followingtable.

TABLE 3.1.1-1 Investment in Return Per Unit States State (′000) if StateOccurs (0, 80] 1,046.58 94.55 (80, 80.5] 870.67 113.85 (80.5, 81]1,411.35 69.85 (81, 81.5] 2,157.85 45.34 (81.5, 82] 3,115.03 31.1 (82,82.5] 4,250.18 22.53 (82.5, 83] 5,486.44 17.23 (83, 83.5] 6,707.18 13.91(83.5, 84] 7,772.68 11.87 (84, 84.5] 8,546.50 10.7 (84.5, 85] 8,924.7110.2 (85, 85.5] 8,858.85 10.29 (85.5, 86] 8,366.06 10.95 (86, 86.5]7,523.13 12.29 (86.5, 87] 6,447.26 14.51 (87, 87.5] 5,270.01 17.98(87.5, 88] 4,112.05 23.31 (88, 88.5] 3,065.21 31.62 (88.5, 89] 2,184.544.78 (89, 89.5] 1,489.58 66.13 (89.5, 90] 972.56 101.82 (90, ∞]1,421.61 69.34

Consistent with the design of a preferred embodiment of a group of DBARcontingent claims, the amount invested for any given state is inverselyrelated to the unit return for that state.

In preferred embodiments of groups of DBAR contingent claims, traderscan invest in none, one or many states. It may be possible in preferredembodiments to allow traders efficiently to invest in a set, subset orcombination of states for the purposes of generating desireddistributions of payouts across the states. In particular, traders maybe interested in replicating payout distributions which are common inthe traditional markets, such as paybuts corresponding to a long stockposition, a short futures position, a long option straddle position, adigital put or digital call option.

If in this Example 3.1.1 a trader desired to hedge his exposure toextreme outcomes in MSFT stock, then the trader could invest in statesat each end of the distribution of possible outcomes. For instance, atrader might decide to invest $100,000 in states encompassing pricesfrom $0 up to and including $83 (i.e., (0,83]) and another $100,000 instates encompassing prices greater than $86.50 (i.e., (86.5,∞]). Thetrader may further desire that no matter what state actually occurswithin these ranges (should the state occur in either range) upon thefulfillment of the predetermined termination criteria, an identicalpayout will result. In this Example 3.1.1, a multi-state investment iseffectively a group of single state investments over each multi-staterange, where an amount is invested in each state in the range inproportion to the amount previously invested in that state. If, forexample, the returns provided in Table 3.1.1-1 represent finalizedprojected returns at the end of the trading period, then eachmulti-state investment may be allocated to its constituent states on apro-rata or proportional basis according to the relative amountsinvested in the constituent states at the close of trading. In this way,more of the multi-state investment is allocated to states with largerinvestments and less allocated to the states with smaller investments.

Other desired payout distributions across the states can be generated byallocating the amount invested among the constituent states in differentways so as achieve a trader's desired payout distribution. A trader mayselect, for example, both the magnitude of the payouts and how thosepayouts are to be distributed should each state occur and let the DBARexchange's multi-state allocation methods determine (1) the size of theamount invested in each particular constituent state; (2) the states inwhich investments will be made, and (3) how much of the total amount tobe invested will be invested in each of the states so determined. Otherexamples below demonstrate how such selections may be implemented.

Since in preferred embodiments the final projected returns are not knownuntil the end of a given trading period, in such embodiments a previousmulti-state investment is reallocated to its constituent statesperiodically as the amounts invested in each state (and thereforereturns) change during the trading period. At the end of the tradingperiod when trading ceases and projected returns are finalized, in apreferred embodiment a final reallocation is made of all the multi-stateinvestments. In preferred embodiments, a suspense account is used torecord and reallocate multi-state investments during the course oftrading and at the end of the trading period.

Referring back to the illustration assuming two multi-state trades overthe ranges (0.83] and (86.5,∞] for MSFT stock, Table 3.1.1-2 shows howthe multi-state investments in the amount of $100,000 each could beallocated according to a preferred embodiment to the individual statesover each range in order to achieve a payout for each multi-state rangewhich is identical regardless of which state occurs within each range.In particular, in this illustration the multi-state investments areallocated in proportion to the previously invested amount in each state,and the multi-state investments marginally lower returns over (0,83] and(86.5,∞], but marginally increase returns over the range (83, 86.5], asexpected.

To show that the allocation in this example has achieved its goal ofdelivering the desired payouts to the trader, two payouts for the (0,83] range are considered. The payout, if constituent state (80.5, 81]occurs, is the amount invested in that state ($7.696) multiplied by oneplus the return per unit if that state occurs, or(1+69.61)*7.696=$543.40. A similar analysis for the state (82.5, 83]shows that, if it occurs, the payout is equal to(1+17.162)*29.918=$543.40. Thus, in this illustration, the traderreceives the same payout no matter which constituent state occurs withinthe multi-state investment. Similar calculations can be performed forthe range [86.5,∞]. For example, under the same assumptions, the payoutfor the constituent state [86.5,87] would receive a payout of $399.80 ifthe stock price fill in that range after the fulfillment of all of thepredetermined termination criteria. In this illustration, eachconstituent state over the range [86.5,∞] would receive a payout of$399.80, no matter which of those states occurs.

TABLE 3.1.1-2 Traded Amount Return Per Multi-State in State Unit ifAllocation States (′000) State Occurs (′000) (0, 80] 1052.29 94.22 5.707(80, 80.5] 875.42 113.46 4.748 (80.5, 81] 1,419.05 69.61 7.696 (81,81.5] 2,169.61 45.18 11.767 (81.5, 82] 3,132.02 30.99 16.987 (82, 82.5]4,273.35 22.45 23.177 (82.5, 83] 5,516.36 17.16 29.918 (83, 83.5]6,707.18 13.94 (83.5, 84] 7,772.68 11.89 (84, 84.5] 8,546.50 10.72(84.5, 85] 8,924.71 10.23 (85, 85.5] 8,858.85 10.31 (85.5, 86] 8,366.0610.98 (86, 86.5] 7,523.13 12.32 (86.5, 87] 6,473.09 14.48 25.828 (87,87.5] 5,291.12 17.94 21.111 (87.5, 88] 4,128.52 23.27 16.473 (88, 88.5]3,077.49 31.56 12.279 (88.5, 89] 2,193.25 44.69 8.751 (89, 89.5]1,495.55 66.00 5.967 (89.5, 90] 976.46 101.62 3.896 (90, ∞] 1,427.3169.20 5.695

Options on equities and equity indices have been one of the moresuccessful innovations in the capital markets. Currently, listed optionsproducts exist for various underlying equity securities and indices andfor various individual option series. Unfortunately, certain marketslack liquidity. Specifically, liquidity is usually limited to only ahandful of the most widely recognized names. Most option markets areessentially dealer-based. Even for options listed on an exchange,market-makers who stand ready to buy or sell options across all strikesand maturities are a necessity. Although market participants trading aparticular option share an interest in only one underlying equity, theexistence of numerous strike prices scatters liquidity coming into themarket thereby making dealer support essential. In all but the mostliquid and active exchange-traded options, chances are rare that twooption orders will meet for the same strike, at the same price, at thesame time, and for the same volume. Moreover, market-makers in listedand over-the-counter (OTC) equities must allocate capital and managerisk for all their positions. Consequently, the absolute amount ofcapital that any one market-maker has on hand is naturally constrainedand may be insufficient to meet the volume of institutional demand.

The utility of equity and equity-index options is further constrained bya lack of transparency in the OTC markets. Investment banks typicallyoffer customized option structures to satisfy their customers.Customers, however, are sometimes hesitant to trade in environmentswhere they have no means of viewing the market and so are uncertainabout getting the best prevailing price.

Groups of DBAR contingent claims can be structured using the system andmethods of the present invention to provide market participants with afuller, more precise view of the price for risks associated with aparticular equity.

Example 3.1.2 Multiple Multi-State Investments

If numerous multi-state investments are made for a group of DBARcontingent claims, then in a preferred embodiment an iterative procedurecan be employed to allocate all of the multi-state investments to theirrespective constituent states. In preferred embodiments, the goal wouldbe to allocate each multi-state investment in response to changes inamounts invested during the trading period, and to make a finalallocation at the end of the trading period so that each multi-stateinvestment generates the payouts desired by the respective trader. Inpreferred embodiments, the process of allocating multi-state investmentscan be iterative, since allocations depend upon the amounts tradedacross the distribution of states at any point in time. As aconsequence, in preferred embodiments, a given distribution of investedamounts will result in a certain allocation of a multi-state investment.When another multi-state investment is allocated, the distribution ofinvested amounts across the defined states may change and thereforenecessitate the reallocation of any previously allocated multi-stateinvestments. In such preferred embodiments, each multi-state allocationis re-performed so that, after a number of iterations through all of thepending multi-state investments, both the amounts invested and theirallocations among constituent states in the multi-state investments nolonger change with each successive iteration and a convergence isachieved. In preferred embodiments, when convergence is achieved,further iteration and reallocation among the multi-state investments donot change any multi-state allocation, and the entire distribution ofamounts invested across the states remains stable and is said to be inequilibrium. Computer code, as illustrated in Table 1 above or relatedcode readily apparent to one of skill in the art, can be used toimplement this iterative procedure.

A simple example demonstrates a preferred embodiment of an iterativeprocedure that may be employed. For purposes of this example, apreferred embodiment of the following assumptions are made: (i) thereare four defined states for the group of DBAR contingent claims; (ii)prior to the allocation of any multi-state investments, $100 has beeninvested in each state so that the unit return for each of the fourstates is 3; (iii) each desires that each constituent state in amulti-state investment provides the same payout regardless of whichconstituent state actually occurs; and (iv) that the following othermulti-state investments have been made:

TABLE 3.1.2-1 Investment Invested Number State 1 State 2 State 3 State 4Amount, $ 1001 X X 0 0 100 1002 X 0 X X 50 1003 X X 0 0 120 1004 X X X 0160 1005 X X X 0 180 1006 0 0 X X 210 1007 X X X 0 80 1008 X 0 X X 9501009 X X X 0 1000 1010 X X 0 X 500 1011 X 0 0 X 250 1012 X X 0 0 1001013 X 0 X 0 500 1014 0 X 0 X 1000 1015 0 X X 0 170 1016 0 X 0 X 1201017 X 0 X 0 1000 1018 0 0 X X 200 1019 X X X 0 250 1020 X X 0 X 3001021 0 X X X 100 1022 X 0 X X 400where an “X” in each state represents a constituent state of themulti-state trade. Thus, as depicted in Table 3.1.2-1, trade number 1001in the first row is a multi-state investment of $100 to be allocatedamong constituent states 1 and 2, trade number 1002 in the second row isanother multi-state investment in the amount of $50 to be allocatedamong constituent states 1, 3, and 4; etc.

Applied to the illustrative multi-state investment described above, theiterative procedure described above and embodied in the illustrativecomputer code in Table 1, results in the following allocations:

TABLE 3.1.2-2 Investment State 1 State 2 State 3 State 4 Number ($) ($)($) ($) 1001 73.8396 26.1604 0 0 1002 26.66782 0 12.53362 10.79856 100388.60752 31.39248 0 0 1004 87.70597 31.07308 41.22096 0 1005 98.6692134.95721 46.37358 0 1006 0 0 112.8081 97.19185 1007 43.85298 15.5365420.61048 0 1008 506.6886 0 238.1387 205.1726 1009 548.1623 194.2067257.631 0 1010 284.2176 100.6946 0 115.0878 1011 177.945 0 0 72.055 101273.8396 26.1604 0 0 1013 340.1383 0 159.8617 0 1014 0 466.6488 0533.3512 1015 0 73.06859 96.93141 0 1016 0 55.99785 0 64.00215 1017680.2766 0 319.7234 0 1018 0 0 107.4363 92.56367 1019 137.0406 48.5516864.40774 0 1020 170.5306 60.41675 0 69.05268 1021 0 28.82243 38.2352932.94229 1022 213.3426 0 100.2689 86.38848In Table 3.1.2-2 each row shows the allocation among the constituentstates of the multi-state investment entered into the corresponding rowof Table 3.1.2-1, the first row of Table 3.1.2-2 that investment number1001 in the amount of $100 has been allocated $73.8396 to state 1 andthe remainder to state 2.

It may be shown that the multi-state allocations identified above resultin payouts to traders which are desired by the traders—that is, in thisexample the desired payouts are the same regardless of which stateoccurs among the constituent states of a given multi-state investment.Based on the total amount invested as reflected in Table 3.1.2-2 andassuming a zero transaction fee, the unit returns for each state are:

State 1 State 2 State 3 State 4 Return Per 1.2292 5.2921 3.7431 4.5052Dollar InvestedConsideration of Investment 1022 in this example, illustrates theuniformity of payouts for each state in which an investment is made(i.e., states 1, 3 and 4). If state 1 occurs, the total payout to thetrader is the unit return for state 1—1.2292—multiplied by the amounttraded for state 1 in trade 1022—$213.3426—plus the initialtrade—$213.3426. This equals 1.2292*213.3426+213.3426=$475.58. If state3 occurs, the payout is equal to 3.7431*100.2689+100.2689=$475.58.Finally, if state 4 occurs, the payout is equal to4.5052*86.38848+86.38848=$475.58. So a preferred embodiment of amulti-state allocation in this example has effected an allocation amongthe constituent states so that (1) the desired payout distributions inthis example are achieved, i.e., payouts to constituent states are thesame no matter which constituent state occurs, and (2) furtherreallocation iterations of multi-state investments do not change therelative amounts invested across the distribution of states for all themulti-state trades.

Example 3.1.3 Alternate Price Distributions

Assumptions regarding the likely distribution of traded amounts for agroup of DBAR contingent claims may be used, for example, to computereturns for each defined state per unit of amount invested at thebeginning of a trading period (“opening returns”). For various reasons,the amount actually invested in each defined state may not reflect theassumptions used to calculate the opening returns. For instance,investors may speculate that the empirical distribution of returns overthe time horizon may differ from the no-arbitrage assumptions typicallyused in option pricing. Instead of a lognormal distribution, moreinvestors might make investments expecting returns to be significantlypositive rather than negative (perhaps expecting favorable news). InExample 3.1.1, for instance, if traders invested more in states above$85 for the price of MSFT common stock, the returns to states below $85could therefore be significantly higher than returns to states above$85.

In addition, it is well known to derivatives traders that traded optionprices indicate that price distributions differ markedly fromtheoretical lognormality or similar theoretical distributions. Theso-called volatility skew or “smile” refers to out-of-the-money put andcall options trading at higher implied volatilities than options closerto the money. This indicates that traders often expect the distributionof prices to have greater frequency or mass at the extreme observationsthan predicted according to lognormal distributions. Frequently, thiseffect is not symmetric so that, for example, the probability of largelower price outcomes are higher than for extreme upward outcomes.Consequently, in a group of DBAR contingent claims of the presentinvention, investment in states in these regions may be more prevalentand, therefore, finalized returns on outcomes in those regions lower.For example, using the basic DBAR contingent claim information fromExample 3.1.1, the following returns may prevail due to investorexpectations of return distributions that have more frequent occurrencesthan those predicted by a lognormal distribution, and thus are skewed tothe lower possible returns. In statistical parlance, such a distributionexhibits higher kurtosis and negative skewness in returns than theillustrative distribution used in Example 3.1.1 and reflected in Table3.1.1-1.

TABLE 3.1.3-1 DBAR Contingent Claim Returns Illustrating NegativelySkewed and Leptokurtotic Return Distribution Amount Invested Return PerUnit States in State (′000) if State Occurs (0, 80] 3,150 30.746 (80,80.5] 1,500 65.667 (80.5, 81] 1,600 61.5 (81, 81.5] 1,750 56.143 (81.5,82] 2,100 46.619 (82, 82.5] 2,550 38.216 (82.5, 83] 3,150 30.746 (83,83.5] 3,250 29.769 (83.5, 84] 3,050 31.787 (84, 84.5] 8,800 10.363(84.5, 85] 14,300 5.993 (85, 85.5] 10,950 8.132 (85.5, 86] 11,300 7.85(86, 86.5] 10,150 8.852 (86.5, 87] 11,400 7.772 (87, 87.5] 4,550 20.978(87.5, 88] 1,350 73.074 (88, 88.5] 1,250 79.0 (88.5, 89] 1,150 85.957(89, 89.5] 700 141.857 (89.5, 90] 650 152.846 (90, ∞] 1,350 73.074

The type of complex distribution illustrated in Table 3.1.3-1 isprevalent in the traditional markets. Derivatives traders, actuaries,risk managers and other traditional market participants typically usesophisticated mathematical and analytical tools in order to estimate thestatistical nature of future distributions of risky market outcomes.These tools often rely on data sets (e.g., historical time series,options data) that may be incomplete or unreliable. An advantage of thesystems and methods of the present invention is that such analyses fromhistorical data need not be complicated, and the full outcomedistribution for a group of DBAR contingent claims based on any givenevent is readily available to all traders and other interested partiesnearly instantaneously after each investment.

Example 3.1.4 States Defined for Return Uniformity

It is also possible in preferred embodiments of the present invention todefine states for a group of DBAR contingent claims with irregular orunevenly distributed intervals, for example, to make the traded amountacross the states more liquid or uniform. States can be constructed froma likely estimate of the final distribution of invested amounts in orderto make the likely invested amounts, and hence the returns for eachstate, as uniform as possible across the distribution of states. Thefollowing table illustrates the freedom, using the event and tradingperiod from Example 3.1.1, to define states so as to promoteequalization of the amount likely to be invested in each state.

TABLE 3.1.4-1 State Definition to Make Likely Demand Uniform AcrossStates Invested Amount Return Per Unit States in State (‘000) if StateOccurs (0, 81.403] 5,000 19 (81.403, 82.181] 5,000 19 (82.181, 82.71]5,000 19 (82.71, 83.132] 5,000 19 (83.132, 83.497] 5,000 19 (83.497,83.826] 5,000 19 (83.826, 84.131] 5,000 19 (84.131, 84.422] 5,000 19(84.422, 84.705] 5,000 19 (84.705, 84.984] 5,000 19 (84.984, 85.264]5,000 19 (85.264, 85.549] 5,000 19 (85.549, 85.845] 5,000 19 (85.845,86.158] 5,000 19 (86.158, 86.497] 5,000 19 (86.497, 86.877] 5,000 19(86.877, 87.321] 5,000 19 (87.321, 87.883] 5,000 19 (87.883, 88.722]5,000 19 (88.722, ∞] 5,000 19

If investor expectations coincide with the often-used assumption of thelognormal distribution, as reflected in this example, then investmentactivity in the group of contingent claims reflected in Table 3.1.4-1will converge to investment of the same amount in each of the 20 statesidentified in the table. Of course, actual trading will likely yieldfinal market returns which deviate from those initially chosen forconvenience using a lognormal distribution.

Example 3.1.5 Government Bond—Uniformly Constructed States

The event, defined states, predetermined termination criteria and otherrelevant data for an illustrative group of DBAR contingent claims basedon a U.S. Treasury Note are set forth below:

-   -   Underlying Security: United States Treasury Note, 5.5%, May 31,        2003    -   Bond Settlement Date: Jun. 25, 1999    -   Bond Maturity Date: May 31, 2003    -   Contingent Claim Expiration: Jul. 2, 1999, Market Close, 4:00        p.m. EST    -   Trading Period Start Date: Jun. 25, 1999, 4:00 p.m., EST    -   Trading Period End Date: Jun. 28, 1999, 4:00 p.m., EST    -   Next Trading Period Open: Jun. 28, 1999, 4:00 p.m., EST    -   Next Trading Period Close Jun. 29, 1999, 4:00 p.m., EST    -   Event: Closing Composite Price as reported on Bloomberg at Claim        Expiration    -   Trading Time: 1 day    -   Duration from TED: 5 days    -   Coupon: 5.5%    -   Payment Frequency Semiannual    -   Daycount Basis Actual/Actual    -   Dividends Payable over Time Horizon: 2.75 per 100 on Jun. 30,        1999    -   Treasury note repo rate over Time Horizon: 4.0% (Actual/360        daycount)    -   Spot Price: 99.8125    -   Forward Price at Expiration: 99.7857    -   Price Volatility: 4.7%    -   Trade and Payout Units: U.S. Dollars    -   Total Demand in Current Trading Period: $50 million    -   Transaction Fee: 25 basis points (0.0025%)

TABLE 3.1.5-1 DBAR Contingent Claims on U.S. Government Note Investmentin Unit Return if States State ($) State Occurs (0, 98] 139690.1635356.04 (98, 98.25] 293571.7323 168.89 (98.25, 98.5] 733769.9011 66.97(98.5, 98.75] 1574439.456 30.68 (98.75, 99] 2903405.925 16.18 (99, 99.1]1627613.865 29.64 (99.1, 99.2] 1914626.631 25.05 (99.2, 99.3]2198593.057 21.68 (99.3, 99.4] 2464704.885 19.24 (99.4, 99.5]2697585.072 17.49 (99.5, 99.6] 2882744.385 16.30 (99.6, 99.7]3008078.286 15.58 (99.7, 99.8] 3065194.576 15.27 (99.8, 99.9]3050276.034 15.35 (99.9, 100] 2964602.039 15.82 (100, 100.1] 2814300.65716.72 (100.1, 100.2] 2609637.195 18.11 (100.2, 100.3] 2363883.036 20.10(100.3, 100.4] 2091890.519 22.84 (100.4, 100.5] 1808629.526 26.58(100.5, 100.75] 3326547.254 13.99 (100.75, 101] 1899755.409 25.25 (101,101.25] 941506.1374 51.97 (101.25, 101.5] 405331.6207 122.05 (101.5, ∞]219622.6373 226.09

This Example 3.1.5 and Table 3.1.5-1 illustrate how readily the methodsand systems of the present invention may be adapted to sources of risk,whether from stocks, bonds, or insurance claims. Table 3.1.5-1 alsoillustrates a distribution of defined states which is irregularlyspaced—in this case finer toward the center of the distribution andcoarser at the ends—in order to increase the amount invested in theextreme states.

Example 3.1.6 Outperformance Asset Allocation—Uniform Range

One of the advantages of the system and methods of the present inventionis the ability to construct groups of DBAR contingent claims based onmultiple events and their inter-relationships. For example, many indexfund money managers often have a fundamental view as to whether indicesof high quality fixed income securities will outperform major equityindices. Such opinions normally are contained within a manager's modelfor allocating funds under management between the major asset classessuch as fixed income securities, equities, and cash.

This Example 3.1.6 illustrates the use of a preferred embodiment of thesystems and methods of the present invention to hedge the real-worldevent that one asset class will outperform another. The illustrativedistribution of investments and calculated opening returns for the groupof contingent claims used in this example are based on the assumptionthat the levels of the relevant asset-class indices are jointlylognormally distributed with an assumed correlation. By defining a groupof DBAR contingent claims on a joint outcome of two underlying events,traders are able to express their views on the co-movements of theunderlying events as captured by the statistical correlation between theevents. In this example, the assumption of a joint lognormaldistribution means that the two underlying events are distributed asfollows:

${\overset{\sim}{V}}_{\theta}^{1} = {\left( {\frac{V_{\tau}^{1}}{Z^{1}\left( {\tau,\theta} \right)} - \frac{D^{1}\left( {\tau,\theta} \right)}{Z^{1}\left( {\tau,\theta} \right)}} \right)*^{{{- \sigma_{1}^{2}}/2}*{({\theta - \tau})}}*^{\sigma_{1}*\sqrt{\theta - \tau}*{dz}_{1}}}$${\overset{\sim}{V}}_{\theta}^{2} = {\left( {\frac{V_{\tau}^{2}}{Z^{2}\left( {\tau,\theta} \right)} - \frac{D^{2}\left( {\tau,\theta} \right)}{Z^{2}\left( {\tau,\theta} \right)}} \right)*^{{{- \sigma_{2}^{2}}/2}*{({\theta - \tau})}}*^{\sigma_{2}*\sqrt{\theta - \tau}*{dz}_{2}}}$${g\left( {{dz}_{1},{dz}_{2}} \right)} = {\frac{1}{2*\pi*\sqrt{1 - \rho^{2}}}*{\exp \left( {- \frac{\left( {{dz}_{1}^{2} + {dz}_{2}^{2} - {2*\rho*{dz}_{1}*{dz}_{1}}} \right)}{2*\left( {1 - \rho^{2}} \right)}} \right)}}$

where the subscripts and superscripts indicate each of the two events,and g(dz₁,dz₂) is the bivariate normal distribution with correlationparameter ρ, and the notation otherwise corresponds to the notation usedin the description above of DBAR Range Derivatives.

The following information includes the indices, the trading periods, thepredetermined termination criteria, the total amount invested and thevalue units used in this Example 3.1.6:

-   -   Asset Class 1: JP Morgan United States Government Bond Index        (“JPMGBI”)    -   Asset Class 1 Forward Price at Observation: 250.0    -   Asset Class 1 Volatility: 5%    -   Asset Class 2: S&P 500 Equity Index (“SP500”)    -   Asset Class 2 Forward Price at Observation: 1410    -   Asset Class 2 Volatility: 18%    -   Correlation Between Asset Classes: 0.5    -   Contingent Claim Expiration: Dec. 31, 1999    -   Trading Start Date: Jun. 30, 1999    -   Current Trading Period Start Date: Jul. 1, 1999    -   Current Trading Period End Date: Jul. 30, 1999    -   Next Trading Period Start Date: Aug. 2, 1999    -   Next Trading Period End Date: Aug. 31, 1999    -   Current Date Jul. 12, 1999    -   Last Trading Period End Date: Dec. 30, 1999    -   Aggregate Investment for Current Trading Period: $100 million    -   Trade and Payout Value Units: U.S. Dollars        Table 3.1.6 shows the illustrative distribution of state returns        over the defined states for the joint outcomes based on this        information, with the defined states as indicated.

TABLE 3.1.6-1 Unit Returns for Joint Performance of S&P 500 and JPMGBIJPMGBI (0, (233, (237, (241, (244, (246, (248, (250, (252, (255, (257,(259, (264, (268, State 233] 237] 241] 244] 246] 248] 250] 252] 255]257] 259] 264] 268] ∞] SP500   (0, 1102] 246 240 197 413 475 591 7981167 1788 3039 3520 2330 11764 18518 (1102, 1174] 240 167 110 197 205230 281 373 538 841 1428 1753 7999 11764 (1174, 1252] 197 110 61 99 9498 110 135 180 259 407 448 1753 5207 (1252, 1292] 413 197 99 145 130 128136 157 197 269 398 407 1428 5813 (1292, 1334] 475 205 94 130 113 106108 120 144 189 269 259 841 3184 (1334, 1377] 591 230 98 128 106 95 9399 115 144 197 180 538 1851 (1377, 1421] 798 281 110 136 108 93 88 89 99120 157 135 373 1167 (1421, 1467] 1167 373 135 157 120 99 89 88 93 108136 110 281 798 (1467, 1515] 1851 538 180 197 144 115 99 93 95 106 12898 230 591 (1515, 1564] 3184 841 259 269 189 144 120 108 106 113 130 94205 475 (1564, 1614] 5813 1428 407 398 269 197 157 136 128 130 145 99197 413 (1614, 1720] 5207 1753 448 407 259 180 135 110 98 94 99 61 110197 (1720, 1834] 11764 7999 1753 1428 841 538 373 281 230 205 197 110167 240 (1834, ∞]   18518 11764 2330 3520 3039 1788 1167 798 591 475 413197 240 246

In Table 3.1.6-1, each cell contains the unit returns to the joint statereflected by the row and column entries. For example, the unit return toinvestments in the state encompassing the joint occurrence of the JPMGBIclosing on expiration at 249 and the SP500 closing at 1380 is 88. Sincethe correlation between two indices in this example is assumed to be0.5, the probability both indices will change in the same direction isgreater that the probability that both indices will change in oppositedirections. In other words, as represented in Table 3.1.6-1, unitreturns to investments in states represented in cells in the upper leftand lower right of the table—i.e., where the indices are changing in thesame direction—are lower, reflecting higher implied probabilities, thanunit returns to investments to states represented in cells in the lowerleft and upper right of Table 3.1.6-1—i.e., where the indices arechanging in opposite directions.

As in the previous examples and in preferred embodiments, the returnsillustrated in Table 3.1.6-1 could be calculated as opening indicativereturns at the start of each trading period based on an estimate of whatthe closing returns for the trading period are likely to be. Theseindicative or opening returns can serve as an “anchor point” forcommencement of trading in a group of DBAR contingent claims. Of course,actual trading and trader expectations may induce substantial departuresfrom these indicative values.

Demand-based markets or auctions can be structured to trade DBARcontingent claims, including, for example, digital options, based onmultiple underlying events or variables and their inter-relationships.Market participants often have views about the joint outcome of twounderlying events or assets. Asset allocation managers, for example, areconcerned with the relative performance of bonds versus equities. Anadditional example of multivariate underlying events follows:

-   -   Joint Performance: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on the joint performance or        observation of two different variables. For example, digital        options traded in a demand-based market or auction can be based        on an underlying event defined as the joint observation of        non-farm payrolls and the unemployment rate.

Example 3.1.7 Corporate Bond Credit Risk

Groups of DBAR contingent claims can also be constructed on creditevents, such as the event that one of the major credit rating agencies(e.g., Standard and Poor's, Moodys) changes the rating for some or allof a corporation's outstanding securities. Indicative returns at theoutset of trading for a group of DBAR contingent claims oriented to acredit event can readily be constructed from publicly available datafrom the rating agencies themselves. For example, Table 3.1.7-1 containsindicative returns for an assumed group of DBAR contingent claims basedon the event that a corporation's Standard and Poor's credit rating fora given security will change over a certain period of time. In thisexample, states are defined using the Standard and Poor's creditcategories, ranging from AAA to D (default). Using the methods of thepresent invention, the indicative returns are calculated usinghistorical data on the frequency of the occurrence of these definedstates. In this example, a transaction fee of 1% is charged against theaggregate amount invested in the group of DBAR contingent claims, whichis assumed to be $100 million.

TABLE 3.1.7-1 Illustrative Returns for Credit DBAR Contingent Claimswith 1% Transaction Fee Indicative Current To New Historical Invested inReturn to Rating Rating Probability State ($) State A− AAA 0.0016160,000 617.75 A− AA+ 0.0004 40,000 2474.00 A− AA 0.0012 120,000 824.00A− AA− 0.003099 309,900 318.46 A− A+ 0.010897 1,089,700 89.85 A− A0.087574 8,757,400 10.30 A− A− 0.772868 77,286,800 0.28 A− BBB+ 0.0689796,897,900 13.35 A− BBB 0.03199 3,199,000 29.95 A− BBB− 0.007398 739,800132.82 A− BB+ 0.002299 229,900 429.62 A− BB 0.004999 499,900 197.04 A−BB− 0.002299 229,900 429.62 A− B+ 0.002699 269,900 365.80 A− B 0.000440,000 2474.00 A− B− 0.0004 40,000 2474.00 A− CCC 1E−04 10,000 9899.00A− D 0.0008 80,000 1236.50

In Table 3.1.7-1, the historical probabilities over the mutuallyexclusive and collectively exhaustive states sum to unity. Asdemonstrated above in this specification, in preferred embodiments, thetransaction fee affects the probability implied for each state from theunit return for that state.

Actual trading is expected almost always to alter illustrativeindicative returns based on historical empirical data. This Example3.1.7 indicates how efficiently groups of DBAR contingent claims can beconstructed for all traders or firms exposed to particular credit riskin order to hedge that risk. For example, in this Example, if a traderhas significant exposure to the A− rated bond issue described above, thetrader could want to hedge the event corresponding to a downgrade byStandard and Poor's. For example, this trader may be particularlyconcerned about a downgrade corresponding to an issuer default or “D”rating. The empirical probabilities suggest a payout of approximately$1,237 for each dollar invested in that state. If this trader has$100,000,000 of the corporate issue in his portfolio and a recovery ofratio of 0.3 can be expected in the event of default, then, in order tohedge $70,000,000 of default risk, the trader might invest in the stateencompassing a “D” outcome. To hedge the entire amount of the defaultrisk in this example, the amount of the investment in this state shouldbe $70,000,000/$1,237 or $56,589.

This represents approximately 5.66 basis points of the trader's positionsize in this bond (i.e., $56,589/$100,000,000=0.00056)] which probablyrepresents a reasonable cost of credit insurance against default. Actualinvestments in this group of DBAR contingent claims could alter thereturn on the “D” event over time and additional insurance might need tobe purchased.

Demand-based markets or auctions can be structured to offer a widevariety of products related to common measures of credit quality,including Moody's and S&P ratings, bankruptcy statistics, and recoveryrates. For example, DBAR contingent claims can be based on an underlyingevent defined as the credit quality of Ford corporate debt as defined bythe Standard & Poor's rating agency.

Example 3.1.8 Economic Statistics

As financial markets have become more sophisticated, statisticalinformation that measures economic activity has assumed increasingimportance as a factor in the investment decisions of marketparticipants. Such economic activity measurements may include, forexample, the following U.S. federal government and U.S. and foreignprivate agency statistics:

-   -   Employment, National Output, and Income (Non-farm Payrolls,        Gross Domestic Product, Personal Income)    -   Orders, Production, and Inventories (Durable Goods Orders,        Industrial Production, Manufacturing Inventories)    -   Retail Sales, Housing Starts, Existing Home Sales, Current        Account Balance, Employment Cost Index, Consumer Price Index,        Federal Funds Target Rate    -   Agricultural statistics released by the U.S.D.A. (crop reports,        etc.)    -   The National Association of Purchasing Management (NAPM) survey        of manufacturing Standard and Poor's Quarterly Operating        Earnings of the S&P 500    -   The semiconductor book-to-bill ratio published by the        Semiconductor Industry Association    -   The Halifax House Price Index used extensively as an        authoritative indicator of house price movements in the U.K.        Because the economy is the primary driver of asset performance,        every investor that takes a position in equities, foreign        exchange, or fixed income will have exposure to economic forces        driving these asset prices, either by accident or design.        Accordingly, market participants expend considerable time and        resources to assemble data, models and forecasts. In turn,        corporations, governments, and financial intermediaries depend        heavily on the economic forecasts to allocate resources and to        make market projections.

To the extent that economic forecasts are inaccurate, inefficiencies andsevere misallocation of resources can result. Unfortunately, traditionalderivatives markets fail to provide market participants with a directmechanism to protect themselves against the adverse consequences offalling demand or rising input prices on a macroeconomic level.Demand-based markets or auctions for economic products, however, providemarket participants with a market price for the risk that a particularmeasure of economic activity will vary from expectations and a tool toproperly hedge the risk. The market participants can trade in a marketor an auction where the event of economic significance is an underlyingmeasure of economic activity (e.g., the VIX index as calculated by theCBOE) or a measured parameter related to the underlying event (e.g., animplied volatility or standard deviation of the VIX index).

For example, traders often hedge inflation risk by trading in bondfutures or, where they exist, inflation-protected floating rate bonds. Agroup of DBAR contingent claims can readily be constructed to allowtraders to express expectations about the distribution of uncertaineconomic statistics measuring, for example, the rate of inflation orother relevant variables. The following information describes such agroup of claims:

-   -   Economic Statistic: United States Non-Farm Payrolls    -   Announcement Date May 31, 1999    -   Last Announcement Date: Apr. 30, 1999    -   Expiration: Announcement Date, May 31, 1999    -   Trading Start Date: May 1, 1999    -   Current Trading Period Start Date: May 10, 1999    -   Current Trading Period End Date: May 14, 1999    -   Current Date May 11, 1999    -   Last Announcement: 128,156 ('000)    -   Source: Bureau of Labor Statistics    -   Consensus Estimate: 130,000 (+1.2%)    -   Aggregate Amount Invested in Current Period: $100 million    -   Transaction Fee: 2.0% of Aggregate Traded amount

Using methods and systems of the present invention, states can bedefined and indicative returns can be constructed from, for example,consensus estimates among economists for this index. These estimates canbe expressed in absolute values or, as illustrated, in Table 3.1.8-1 inpercentage changes from the last observation as follows:

TABLE 3.1.8-1 Illustrative Returns For Non-Farm Payrolls Release with 2%Transaction Fee % Chg. Investment Implied In Index in State State State(‘000) State Returns Probability [−100, −5] 100 979 0.001 (−5, −3] 200489 0.002 (−3, −1] 400 244 0.004 (−1, −.5] 500 195 0.005 (−.5, 0] 100097 0.01 (0, .5] 2000 48 0.02 (.5, .7] 3000 31.66667 0.03 (.7, .8] 400023.5 0.04 (.8, .9] 5000 18.6 0.05 (.9, 1.0] 10000 8.8 0.1 (1.0, 1.1]14000 6 0.14 (1.1, 1.2] 22000 3.454545 0.22 (1.2, 1.25] 18000 4.4444440.18 (1.25, 1.3] 9000 9.888889 0.09 (1.3, 1.35] 6000 15.33333 0.06(1.35, 1.40] 3000 31.66667 0.03 (1.40, 1.45] 200 489 0.002 (1.45, 1.5]600 162.3333 0.006 (1.5, 1.6] 400 244 0.004 (1.6, 1.7] 100 979 0.001(1.7, 1.8] 80 1224 0.0008 (1.8, 1.9] 59 1660.017 0.00059 (1.9, 2.0] 591660.017 0.00059 (2.0, 2.1] 59 1660.017 0.00059 (2.1, 2.2] 59 1660.0170.00059 (2.2, 2.4] 59 1660.017 0.00059 (2.4, 2.6] 59 1660.017 0.00059(2.6, 3.0] 59 1660.017 0.00059 (3.0, ∞] 7 13999 0.00007As in examples, actual trading prior to the trading end date would beexpected to adjust returns according to the amounts invested in eachstate and the total amount invested for all the states.

Demand-based markets or auctions can be structured to offer a widevariety of products related to commonly observed indices and statisticsrelated to economic activity and released or published by governments,and by domestic, foreign and international government or privatecompanies, institutions, agencies or other entities. These may include alarge number of statistics that measure the performance of the economy,such as employment, national income, inventories, consumer spending,etc., in addition to measures of real property and other economicactivity. An additional example follows:

-   -   Private Economic Indices & Statistics: Demand-based markets or        auctions can be structured to trade DBAR contingent claims,        including, for example, digital options, based on economic        statistics released or published by private sources. For        example, DBAR contingent claims can be based on an underlying        event defined as the NAPM Index published by the National        Association of Purchasing Managers.        -   Alternative private indices might also include measures of            real property. For example, DBAR contingent claims,            including, for example, digital options, can be based on an            underlying event defined as the level of the Halifax House            Price Index at year-end, 2001.

In addition to the general advantages of the demand-based tradingsystem, demand-based products on economic statistics will provide thefollowing new opportunities for trading and risk management:

-   (1) Insuring against the event risk component of asset price    movements. Statistical releases can often cause extreme short-term    price movements in the fixed income and equity markets. Many market    participants have strong views on particular economic reports, and    try to capitalize on such views by taking positions in the bond or    equity markets. Demand-based markets or auctions on economic    statistics provide participants with a means of taking a direct view    on economic variables, rather than the indirect approach employed    currently.-   (2) Risk management for real economic activity. State governments,    municipalities, insurance companies, and corporations may all have a    strong interest in a particular measure of real economic activity.    For example, the Department of Energy publishes the Electric Power    Monthly which provides electricity statistics at the State, Census    division, and U.S. levels for net generation, fossil fuel    consumption and stocks, quantity and quality of fossil fuels, cost    of fossil fuels, electricity retail sales, associated revenue, and    average revenue. Demand-based markets or auctions based on one or    more of these energy benchmarks can serve as invaluable risk    management mechanisms for corporations and governments seeking to    manage the increasingly uncertain outlook for electric power.-   (3) Sector-specific risk management. The Health Care CPI (Consumer    Price Index) published by the U.S. Bureau of Labor Statistics tracks    the CPI of medical care on a monthly basis in the CPI Detailed    Report. A demand-based market or auction on this statistic would    have broad applicability for insurance companies, drug companies,    hospitals, and many other participants in the health care industry.    Similarly, the semiconductor book-to-bill ratio serves as a direct    measure of activity in the semiconductor equipment manufacturing    industry. The ratio reports both shipments and new bookings with a    short time lag, and hence is a useful measure of supply and demand    balance in the semiconductor industry. Not only would manufacturers    and consumers of semiconductors have a direct financial interest,    but the ratio's status as a bellwether of the general technology    market would invite participation from financial market participants    as well.

Example 3.1.9 Corporate Events

Corporate actions and announcements are further examples of events ofeconomic significance which are usually unhedgable or uninsurable intraditional markets but which can be effectively structured into groupsof DBAR contingent claims according to the present invention.

In recent years, corporate earnings expectations, which are typicallyannounced on a quarterly basis for publicly traded companies, haveassumed increasing importance as more companies forego dividends toreinvest in continuing operations. Without dividends, the present valueof an equity becomes entirely dependent on revenues and earnings streamsthat extend well into the future, causing the equity itself to take onthe characteristics of an option. As expectations of future cash flowschange, the impact on pricing can be dramatic, causing stock prices inmany cases to exhibit option-like behavior.

Traditionally, market participants expend considerable time andresources to assemble data, models and forecasts. To the extent thatforecasts are inaccurate, inefficiencies and severe misallocation ofresources can result. Unfortunately, traditional derivatives marketsfail to provide market participants with a direct mechanism to managethe unsystematic risks of equity ownership. Demand-based markets orauctions for corporate earnings and revenues, however, provide marketparticipants with a concrete price for the risk that earnings andrevenues may vary from expectations and permit them to insure or hedgeor speculate on the risk.

Many data services, such as IBES and FirstCall, currently publishestimates by analysts and a consensus estimate in advance of quarterlyearnings announcements. Such estimates can form the basis for indicativeopening returns at the commencement of trading in a demand-based marketor auction as illustrated below. For this example, a transaction fee ofzero is assumed.

-   -   Underlying security: IBM    -   Earnings Announcement Date: Jul. 21, 1999    -   Consensus Estimate: 0.879/share    -   Expiration: Announcement, Jul. 21, 1999    -   First Trading Period Start Date: Apr. 19, 1999    -   First Trading Period End Date May 19, 1999    -   Current Trading Period Start Date: Jul. 6, 1999    -   Current Trading Period End Date: Jul. 9, 1999    -   Next Trading Period Start Date: Jul. 9, 1999    -   Next Trading Period End Date: Jul. 16, 1999    -   Total Amount Invested in Current Trading Period: $100 million

TABLE 3.1.9-1 Illustrative Returns For IBM Earnings AnnouncementInvested Implied Earnings in State State State0 (‘000 $) Unit ReturnsProbability (−∞, .5] 70 1,427.57 0.0007 (.5, .6] 360 276.78 0.0036 (.6,.65] 730 135.99 0.0073 (.65, .7] 1450 67.97 0.0145 (.7, .74] 2180 44.870.0218 (.74, .78] 3630 26.55 0.0363 (.78, ..8] 4360 21.94 0.0436 (.8,.82] 5820 16.18 0.0582 (.82, .84] 7270 12.76 0.0727 (.84, .86] 872010.47 0.0872 (.86, .87] 10900 8.17 0.109 (.87, .88] 18170 4.50 0.1817(.88, .89] 8720 10.47 0.0872 (.89, .9] 7270 12.76 0.0727 (.9, .91] 509018.65 0.0509 (.91, .92] 3630 26.55 0.0363 (.92, .93] 2910 33.36 0.0291(.93, .95] 2180 44.87 0.0218 (.95, .97] 1450 67.97 0.0145 (.97, .99]1310 75.34 0.0131 (.99, 1.1] 1160 85.21 0.0116 (1.1, 1.3] 1020 97.040.0102 (1.3, 1.5] 730 135.99 0.0073 (1.5, 1.7] 360 276.78 0.0036 (1.7,1.9] 220 453.55 0.0022 (1.9, 2.1] 150 665.67 0.0015 (2.1, 2.3] 701,427.57 0.0007 (2.3, 2.5] 40 2,499.00 0.0004 (2.5, ∞] 30 3,332.330.0003Consistent with the consensus estimate, the state with the largestinvestment encompasses the range (0.87, 0.88].

TABLE 3.1.9-2 Illustrative Returns for Microsoft Earnings AnnouncementStrike Bid Offer Payout Volume Calls <40 0.9525 0.9575 1.0471 4,100,000<41 0.9025 0.9075 1.1050 1,000,000 <42 0.8373 0.8423 1.1908 9,700 <430.7475 0.7525 1.3333 3,596,700 <44 0.622 0.627 1.6013 2,000,000 <450.4975 0.5025 2.0000 6,000,000 <46 0.3675 0.3725 2.7027 2,500,000 <470.2175 0.2225 4.5455 1,000,000 <48 0.1245 0.1295 7.8740 800,000 <490.086 0.091 11.2994 — <50 0.0475 0.0525 20.000 194,700 Puts <40 0.04250.0475 22.2222 193,100 <41 0.0925 0.0975 10.5263 105,500 <42 0.15770.1627 6.2422 — <43 0.2475 0.2525 4.0000 1,200,000 <44 0.3730 0.37802.6631 1,202,500 <45 0.4975 0.5025 2.0000 6,000,000 <46 0.6275 0.63251.5873 4,256,600 <47 0.7775 0.7825 1.2821 3,545,700 <48 0.8705 0.87551.1455 5,500,000 <49 0.9090 0.9140 1.0971 — <50 0.9475 0.9525 1.05263,700,000

The table above provides a sample distribution of trades that might bemade for an April 23 auction period for Microsoft Q4 corporate earnings(June 2001), due to be released on Jul. 16, 2001.

For example, at 29 times trailing earnings and 28 times consensus 2002earnings, Microsoft is experiencing single digit profit growth and isthe object of uncertainty with respect to sales of Microsoft Office,adoption rates of Windows 2000, and the .Net initiative. In the sampledemand-based market or auction based on earnings expectations depictedabove, a market participant can engage, for example, in the followingtrading tactics and strategies with respect to DBAR digital options.

-   -   A fund manager wishing to avoid market risk at the current time        but who still wants exposure to Microsoft can buy the 0.43        Earnings per Share Call (consensus currently 0.44-45) with        reasonable confidence that reported earnings will be 43 cents or        higher. Should Microsoft report earnings as expected, the trader        earns approximately 33% on invested demand-based trading digital        option premium (i.e., 1/option price of 0.7525). Conversely,        should Microsoft report earnings below 43 cents, the invested        premium would be lost, but the consequences for Microsoft's        stock price would likely be dramatic.    -   A more aggressive strategy would involve selling or        underweighting Microsoft stock, while purchasing a string of        digital options on higher than expected EPS growth. In this        case, the trader expects a multiple contraction to occur over        the short to medium term, as the valuation becomes        unsustainable. Using the market for DBAR contingent claims on        earnings depicted above, a trader with a $5 million notional        exposure to Microsoft can buy a string of digital call options,        as follows:

Strike Premium Price Net Payout .46 $37,000 0.3725 $62,329 .47 22,0000.2225 139,205 .48 6,350 0.1295 181,890 .49 4,425 0.0910 226,091 .50 00.0525 226,091

-   -   -   The payouts displayed immediately above are net of premium            investment. Premiums invested are based on the trader's            assessment of likely stock price (and price multiple)            reaction to a possible earnings surprise. Similar trades in            digital options on earnings would be made in successive            quarters, resulting in a string of options on higher than            expected earnings growth, to protect against an upward shift            in the earnings expectation curve, as shown in FIG. 21.        -   The total cost, for this quarter, amounts to $69,775, just            above a single quarter's interest income on the notional            $5,000,000, invested at 5%.

    -   A trader with a view on a range of earnings expectations for the        quarter can profit from a spread strategy over the distribution.        By purchasing the 0.42 call and selling the 0.46 call, the        trader can construct a digital option spread priced at:        0.8423-0.3675=0.4748. This spread would, consequently, pay out:        1/0.4748=2.106, for every dollar invested.

Many trades can be constructed using demand-based trading for DBARcontingent claims, including, for example, digital options, based oncorporate earnings. The examples shown here are intended to berepresentative, not definitive. Moreover, demand-based trading productscan be based on corporate accounting measures, including a wide varietyof generally accepted accounting information from corporate balancesheets, income statements, and other measures of cash flow, such asearnings before interest, taxes, depreciation, and amortization(EBITDA). The following examples provide a further representativesampling:

-   -   Revenues: Demand-based markets or auctions for DBAR contingent        claims, including, for example, digital options can be based on        a measure or parameter related to Cisco revenues, such as the        gross revenues reported by the Cisco Corporation. The underlying        event for these claims is the quarterly or annual gross revenue        figure for Cisco as calculated and released to the public by the        reporting company.    -   EBITDA (Earnings Before Interest, Taxes, Depreciation,        Amortization): Demand-based markets or auctions for DBAR        contingent claims, including, for example, digital options can        be based on a measure or parameter related to AOL EBITDA, such        as the EBITDA figure reported by AOL that is used to provide a        measure of operating earnings. The underlying event for these        claims is the quarterly or annual EBITDA figure for AOL as        calculated and released to the public by the reporting company.        In addition to the general advantages of the demand-based        trading system, products based on corporate earnings and        revenues may provide the following new opportunities for trading        and risk management:

-   (1) Trading the price of a stock relative to its earnings. Traders    can use a market for earnings to create a “Multiple Trade,” in which    a stock would be sold (or ‘not owned’) and a string of DBAR    contingent claims, including, for example, digital options, based on    quarterly earnings can be used as a hedge or insurance for stock    believed to be overpriced. Market expectations for a company's    earnings may be faulty, and may threaten the stability of a stock    price, post announcement. Corporate announcements that reduce    expectation for earnings and earnings growth highlight the    consequences for high-multiple growth stocks that fail to meet    expectations. For example, an equity investment manager might decide    to underweight a high-multiple stock against a benchmark, and    replace it with a series of DBAR digital options corresponding to a    projected profile for earnings growth. The manager can compare the    cost of this strategy with the risk of owning the underlying    security, based on the company's PE ratio or some other metric    chosen by the fund manager. Conversely, an investor who expects a    multiple expansion for a given stock would purchase demand-based    trading digital put options on earnings, retaining the stock for a    multiple expansion while protecting against a shortfall in reported    earnings.

-   (2) Insuring against an earnings shortfall, while maintaining a    stock position during a period when equity options are deemed too    expensive. While DBAR contingent claims, including, for example,    digital options, based on earnings are not designed to hedge stock    prices, they can provide a cost-effective means to mitigate the risk    of equity ownership over longer term horizons. For example,    periodically, three-month stock options that are slightly    out-of-the-money can command premiums of 10% or more. The ability to    insure against possible earnings or revenue shortfalls one quarter    or more in the future via purchases of DBAR digital options may    represent an attractive alternative to conventional hedge strategies    for equity price risks.

-   (3) Insuring against an earnings shortfall that may trigger credit    downgrades. Fixed income managers worried about potential exposure    to credit downgrades from reduced corporate earnings can use DBAR    contingent claims, including, for example, digital options, to    protect against earnings shortfalls that would impact EBITDA and    prompt declines in corporate bond prices. Conventional fixed income    and convertible bond managers can protect against equity exposures    without a short sale of the corresponding equity shares.

-   (4) Obtaining low-risk, incremental returns. Market participants can    use deep-in-the-money DBAR contingent claims, including, for    example, digital options, based on earnings as a source of low-risk,    uncorrelated returns.

Example 3.1.10 Real Assets

Another advantage of the methods and systems of the present invention isthe ability to structure liquid claims on illiquid underlying assetssuch a real estate. As previously discussed, traditional derivativesmarkets customarily use a liquid underlying market in order to functionproperly. With a group of DBAR contingent claims all that is usuallyrequired is a real-world, observable event of economic significance. Forexample, the creation of contingent claims tied to real assets has beenattempted at some financial institutions over the last several years.These efforts have not been credited with an appreciable impact,apparently because of the primary liquidity constraints inherent in theunderlying real assets.

A group of DBAR contingent claims according to the present invention canbe constructed based on an observable event related to real estate. Therelevant information for an illustrative group of such claims is asfollows:

-   -   Real Asset Index: Colliers ABR Manhattan Office Rent Rates    -   Bloomberg Ticker: COLAMANR    -   Update Frequency Monthly    -   Source: Colliers ABR, Inc.    -   Announcement Date Jul. 31, 1999    -   Last Announcement Date: Jun. 30, 1999    -   Last Index Value: $45.39/sq. ft.    -   Consensus Estimate: $45.50    -   Expiration: Announcement Jul. 31, 1999    -   Current Trading Period Start: Jun. 30, 1999    -   Current Trading Period End: Jul. 7, 1999    -   Next Trading Period Start Jul. 7, 1999    -   Next Trading Period End Jul. 14, 1999

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be calculated or willemerge from actual trader investments according to the methods of thepresent invention as illustrated in Examples 3.1.1-3.1.9.

Demand-based markets or auctions can be structured to offer a widevariety of products related to real assets, such as real estate,bandwidth, wireless spectrum capacity, or computer memory. An additionalexample follows:

-   -   Computer Memory: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on computer memory components.        For example, DBAR contingent claims can be based on an        underlying event defined as the 64 Mb (8×8) PC 133 DRAM memory        chip prices and on the rolling 90-day average of Dynamic Random        Access Memory DRAM prices as reported each Friday by ICIS-LOR, a        commodity price monitoring group based in London.

Example 3.1.11 Energy Supply Chain

A group of DBAR contingent claims can also be constructed using themethods and systems of the present invention to provide hedging vehicleson non-tradable quantities of great economic significance within thesupply chain of a given industry. An example of such an application isthe number of oil rigs currently deployed in domestic U.S. oilproduction. The rig count tends to be a slowly adjusting quantity thatis sensitive to energy prices. Thus, appropriately structured groups ofDBAR contingent claims based on rig counts could enable suppliers,producers and drillers to hedge exposure to sudden changes in energyprices and could provide a valuable risk-sharing device.

For example, a group of DBAR contingent claims depending on the rigcount could be constructed according to the present invention using thefollowing information (e.g., data source, termination criteria, etc).

-   -   Asset Index: Baker Hughes Rig Count U.S. Total    -   Bloomberg Ticker: BAKETOT    -   Frequency: Weekly    -   Source: Baker Hughes, Inc.    -   Announcement Date: Jul. 16, 1999    -   Last Announcement Date: Jul. 9, 1999    -   Expiration Date: Jul. 16, 1999    -   Trading Start Date: Jul. 9, 1999    -   Trading End Date: Jul. 15, 1999    -   Last: 570    -   Consensus Estimate: 580

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9. A varietyof embodiments of DBAR contingent claims, including for example, digitaloptions, can be based on an underlying event defined as the Baker HughesRig Count observed on a semi-annual basis.

Demand-based markets or auctions can be structured to offer a widevariety of products related to power and emissions, includingelectricity prices, loads, degree-days, water supply, and pollutioncredits. The following examples provide a further representativesampling:

-   -   Electricity Prices Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on the price of electricity at        various points on the electricity grid. For example, DBAR        contingent claims can be based on an underlying event defined as        the weekly average price of electricity in kilowatt-hours at the        New York Independent System Operator (NYISO).    -   Transmission Load Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on the actual load (power        demand) experienced for a particular power pool, allowing        participants to trade volume, in addition to price. For example,        DBAR contingent claims can be based on an underlying event        defined as the weekly total load demand experienced by        Pennsylvania-New Jersey-Maryland Interconnect (PJM Western Hub).    -   Water: Demand-based markets or auctions can be structured to        trade DBAR contingent claims, including, for example, digital        options, based on water supply. Water measures are useful to a        broad variety of constituents, including power companies,        agricultural producers, and municipalities. For example, DBAR        contingent claims can be based on an underlying event defined as        the cumulative precipitation observed at weather stations        maintained by the National Weather Service in the Northwest        catchment area, including Washington, Idaho, Montana, and        Wyoming.    -   Emission Allowances: Demand-based markets or auctions can be        structured to trade DBAR contingent claims; including, for        example, digital options, based on emission allowances for        various pollutants. For example, DBAR contingent claims can be        based on an underlying event defined as price of Environmental        Protection Agency (EPA) sulfur dioxide allowances at the annual        market or auction administered by the Chicago Board of Trade.

Example 3.1.12 Mortgage Prepayment Risk

Real estate mortgages comprise an extremely large fixed income assetclass with hundreds of billions in market capitalization. Marketparticipants generally understand that these mortgage-backed securitiesare subject to interest rate risk and the risk that borrowers mayexercise their options to refinance their mortgages or otherwise“prepay” their existing mortgage loans. The owner of a mortgagesecurity, therefore, bears the risk of being “called” out of itsposition when mortgage interest rate levels decline.

Market participants expend considerable time and resources assemblingeconometric models and synthesizing various data populations in order togenerate prepayment projections. To the extent that economic forecastsare inaccurate, inefficiencies and severe misallocation of resources canresult. Unfortunately, traditional derivatives markets fail to providemarket participants with a direct mechanism to protect themselvesagainst a homeowner's exercise of its prepayment option. Demand-basedmarkets or auctions for mortgage prepayment products, however, providemarket participants with a concrete price for prepayment risk.

Groups of DBAR contingent claims can be structured according to thepresent invention, for example, based on the following information:

-   -   Asset Index: FNMA Conventional 30 year One-Month Historical        Aggregate Prepayments    -   Coupon: 6.5%    -   Frequency: Monthly    -   Source: Bloomberg    -   Announcement Date Aug. 1, 1999    -   Last Announcement Date: Jul. 1, 1999    -   Expiration: Announcement Date, Aug. 1, 1999    -   Current Trading Period Start Date: Jul. 1, 1999    -   Current Trading Period End Date: Jul. 9, 1999    -   Last: 303 Public Securities Association Prepayment Speed (“PSA”)    -   Consensus Estimate: 310 PSA

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9.

In addition to the general advantages of the demand-based tradingsystem, products on mortgage prepayments may provide the followingexemplary new opportunities for trading and risk management:

-   (1) Asset-specific applications. In the simplest form, the owner of    a prepayable mortgage-backed security carries, by definition, a    series of short option positions embedded in the asset, whereas a    DBAR contingent claim, including, for example, a digital option,    based on mortgage prepayments would constitute a long option    position. A security owner would have the opportunity to compare the    digital option's expected return with the prospective loss of    principal, correlate the offsetting options, and invest accordingly.    While this tactic would not eliminate reinvestment risks, per se, it    would generate incremental investment returns that would reduce the    security owner's embedded liabilities with respect to short option    positions.-   (2) Portfolio applications. Certainly, a similar strategy could be    applied on an expanded basis to a portfolio of mortgage-backed    securities, or a portfolio of whole mortgage loans.-   (3) Enhancements to specific pools. Certain pools of seasoned    mortgage loans exhibit consistent prepayment patterns, based upon    comprehensible factors—origination period, underwriting standards,    borrower circumstances, geographic phenomena, etc. Because of    homogeneous prepayment performance, mortgage market participants can    obtain greater confidence with respect to the accuracy of    predictions for prepayments in these pools, than in the case of    pools of heterogeneous, newly originated loans that lack a    prepayment history. Market conventions tend to assign lower    volatility estimates to the correlation of prepayment changes in    seasoned pools for given interest rate changes, than in the case of    newer pools. A relatively consistent prepayment pattern for seasoned    mortgage loan pools would heighten the certainty of correctly    anticipating future prepayments, which would heighten the likelihood    of consistent success in trading in DBAR contingent claims such as,    for example, digital options, based on respective mortgage    prepayments. Such digital option investments, combined with seasoned    pools, would tend to enhance annuity-like cash profiles, and reduce    investment risks.-   (4) Prepayment puts plus discount MBS. Discount mortgage-backed    securities tend to enjoy two-fold benefits as interest rates decline    in the form of positive price changes and increases in prepayment    speeds. Converse penalties apply in events of increases in interest    rates, where a discount MBS suffers from adverse price change, and a    decline in prepayment income. A discount MBS owner could offset    diminished prepayment income by investing in DBAR contingent claims,    such as, for example, digital put options, or digital put option    spreads on prepayments. An analogous strategy would apply to    principal-only mortgage-backed securities.-   (5) Prepayment calls plus premium MBS. An expectation of interest    rate declines that accelerate prepayment activity for premium    mortgage-backed securities would motivate a premium bond-holder to    purchase DBAR contingent claims, such as, for example, digital call    options, based on mortgage prepayments to offset losses attributable    to unwelcome paydowns. The analogue would also apply to    interest-only mortgage-backed securities.-   (6) Convexity additions. An investment in a DBAR contingent claim,    such as, for example, a digital option, based on mortgage    prepayments should effectively add convexity to an interest rate    sensitive investment. According to this reasoning, dollar-weighted    purchases of a demand-based market or auction on mortgage    prepayments would tend to offset the negative convexity exhibited by    mortgage-backed securities. It is likely that expert participants in    the mortgage marketplace will analyze and test, and ultimately    harvest, the fruitful opportunities for combinations of DBAR    contingent claims, including, for example, digital options, based on    mortgage prepayments with mortgage-backed securities and    derivatives.

Example 3.1.13 Insurance Industry Loss Warranty (“ILW”)

The cumulative impact of catastrophic and non-catastrophic insurancelosses over the past two years has reduced the capital available in theretrocession market (i.e. reinsurance for reinsurance companies) andpushed up insurance and reinsurance rates for property catastrophecoverage. Because large reinsurance companies operate global businesseswith global exposures, severe losses from catastrophes in one countrytend to drive up insurance and reinsurance rates for unrelated perils inother countries simply due to capital constraints.

As capital becomes scarce and insurance rates increase, marketparticipants usually access the capital markets by purchasingcatastrophic bonds (CAT bonds) issued by special purpose reinsurancecompanies. The capital markets can absorb the risk of loss associatedwith larger disasters, whereas a single insurer or even a group ofinsurers cannot, because the risk is spread across many more marketparticipants.

Unlike traditional capital markets that generally exhibit a naturaltwo-way order flow, insurance markets typically exhibit one-way demandgenerated by participants desiring protection from adverse outcomes.Because demand-based trading products do not require an underlyingsource of supply, such products provide an attractive alternative foraccess to capital.

Groups of DBAR contingent claims can be structured using the system andmethods of the present invention to provide insurance and reinsurancefacilities for property and casualty, life, health and other traditionallines of insurance. The following information provides information tostructure a group of DBAR contingent claims related to large propertylosses from hurricane damage:

-   -   Event: PCS Eastern Excess $5 billion Index    -   Source: Property Claim Services (PCS)    -   Frequency: Monthly    -   Announcement Date Oct. 1, 1999    -   Last Announcement Date: Jul. 1, 1999    -   Last Index Value: No events    -   Consensus Estimate: $1 billion (claims excess of $5 billion)    -   Expiration: Announcement Date, Oct. 1, 1999    -   Trading Period Start Date: Jul. 1, 1999    -   Trading Period End Date: Sep. 30, 1999

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9.

In preferred embodiments of groups of DBAR contingent claims related toproperty-casualty catastrophe losses, the frequency of claims and thedistributions of the severity of losses are assumed and convolutions areperformed in order to post indicative returns over the distribution ofdefined states. This can be done, for example, using compoundfrequency-severity models, such as the Poisson-Pareto model, familiar tothose of skill in the art, which predict, with greater probability thana normal distribution, when losses will be extreme. As indicatedpreviously, in preferred embodiments market activity is expected toalter the posted indicative returns, which serve as informative levelsat the commencement of trading.

Demand-based markets or auctions can be structured to offer a widevariety of products related to insurance industry loss warranties andother insurable risks, including property and non-property catastrophe,mortality rates, mass torts, etc. An additional example follows:

-   -   Property Catastrophe: Demand-based markets or auctions can be        based on the outcome of natural catastrophes, including        earthquake, fire, atmospheric peril, and flooding, etc.        Underlying events can be based on hazard parameters. For        example, DBAR contingent claims can be based on an underlying        event defined as the cumulative losses sustained in California        as the result of earthquake damage in the year 2002, as        calculated by the Property Claims Service (PCS).

In addition to the general advantages of the demand-based tradingsystem, products on catastrophe risk will provide the following newopportunities for trading and risk management:

-   (1) Greater transaction efficiency and precision. A demand-based    trading catastrophe risk product, such as, for example, a DBAR    digital option, allows participants to buy or sell a precise    notional quantity of desired risk, at any point along a catastrophe    risk probability curve, with a limit price for the risk. A series of    loss triggers can be created for catastrophic events that offer    greater flexibility and customization for insurance transactions, in    addition to indicative pricing for all trigger levels. Segments of    risk coverage can be traded with ease and precision. Participants in    demand-based trading catastrophe risk products gain the ability to    adjust risk protection or exposure to a desired level. For example,    a reinsurance company may wish to purchase protection at the tail of    a distribution, for unlikely but extremely catastrophic losses,    while writing insurance in other parts of the distribution where    returns may appear attractive.-   (2) Credit quality. Claims-paying ability of an insurer or reinsurer    represents an important concern for many market participants.    Participants in a demand-based market or auction do not depend on    the credit quality of an individual insurance or reinsurance    company. A demand-based market or auction is by nature self-funding,    meaning that catastrophic losses in other product or geographic    areas will not impair the ability of a demand-based trading    catastrophe risk product to make capital distributions.

Example 3.1.14 Conditional Events

As discussed above, advantage of the systems and methods of the presentinvention is the ability to construct groups of DBAR contingent claimsrelated to events of economic significance for which there is greatinterest in insurance and hedging, but which are not readily hedged orinsured in traditional capital and insurance markets. Another example ofsuch an event is one that occurs only when some related event haspreviously occurred. For purposes of illustration, these two events maybe denoted A and B.

${q{\langle{AB}\rangle}} = \frac{q\left( {A\bigcap B} \right)}{q(B)}$

where q denotes the probability of a state, q

A|B

represents the conditional probability of state A given the prioroccurrence of state and B, and q(A∩B) represents the occurrence of bothstates A and B.

For example, a group of DBAR contingent claims may be constructed tocombine elements of “key person” insurance and the performance of thestock price of the company managed by the key person. Many firms aremanaged by people whom capital markets perceive as indispensable orparticularly important, such as Warren Buffett of Berkshire Hathaway.The holders of Berkshire Hathaway stock have no ready way of insuringagainst the sudden change in management of Berkshire, either due to acorporate action such as a takeover or to the death or disability ofWarren Buffett. A group of conditional DBAR contingent claims can beconstructed according to the present invention where the defined statesreflect the stock price of Berkshire Hathaway conditional on WarrenBuffet's leaving the firm's management. Other conditional DBARcontingent claims that could attract significant amounts for investmentcan be constructed using the methods and systems of the presentinvention, as apparent to one of skill in the art.

Example 3.1.15 Securitization Using a DBAR Contingent Claim Mechanism

The systems and methods of the present invention can also be adapted bya financial intermediary or issuer for the issuance of securities suchas bonds, common or preferred stock, or other types of financialinstruments. The process of creating new opportunities for hedgingunderlying events through the creation of new securities is known as“securitization,” and is also discussed in an embodiment presented inSection 10. Well-known examples of securitization include the mortgageand asset-backed securities markets, in which portfolios of financialrisk are aggregated and then recombined into new sources of financialrisk. The systems and methods of the present invention can be usedwithin the securitization process by creating securities, or portfoliosof securities, whose risk, in whole or part, is tied to an associated orembedded group of DBAR contingent claims. In a preferred embodiment, agroup of DBAR contingent claims is associated with a security much likeoptions are currently associated with bonds in order to create callableand putable bonds in the traditional markets.

This example illustrates how a group of DBAR contingent claims accordingto the present invention can be tied to the issuance of a security inorder to share risk associated with an identified future event among thesecurity holders. In this example, the security is a fixed income bondwith an embedded group of DBAR contingent claims whose value depends onthe possible values for hurricane losses over some time period for somegeographic region.

-   -   Issuer: Tokyo Fire and Marine    -   UnderWriter: Goldman Sachs    -   DBAR Event: Total Losses on a Saffir-Simpson Category 4        Hurricane    -   Geographic: Property Claims Services Eastern North America    -   Date: Jul. 1, 1999-Nov. 1, 1999    -   Size of Issue: 500 million USD.    -   Issue Date Jun. 1, 1999    -   DBAR Trading Period: Jun. 1, 1999-Jul. 1, 1999

In this example, the underwriter Goldman Sachs issues the bond, andholders of the issued bond put bond principal at risk over the entiredistribution of amounts of Category 4 losses for the event. Ranges ofpossible losses comprise the defined states for the embedded group ofDBAR contingent claims. In a preferred embodiment, the underwriter isresponsible for updating the returns to investments in the variousstates, monitoring credit risk, and clearing and settling, andvalidating the amount of the losses. When the event is determined anduncertainty is resolved, Goldman is “put” or collects the bond principalat risk from the unsuccessful investments and allocates these amounts tothe successful investments. The mechanism in this illustration thusincludes:

-   -   (1) An underwriter or intermediary which implements the        mechanism, and    -   (2) A group of DBAR contingent claims directly tied to a        security or issue (such as the catastrophe bond above).

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9.

Example 3.1.16 Exotic Derivatives

The securities and derivatives communities frequently use the term“exotic derivatives” to refer to derivatives whose values are linked toa security, asset, financial product or source of financial risk in amore complicated fashion than traditional derivatives such as futures,call options, and convertible bonds. Examples of exotic derivativesinclude American options, Asian options, barrier options, Bermudanoptions, chooser and compound options, binary or digital options,lookback options, automatic and flexible caps and floors, and shoutoptions.

Many types of exotic options are currently traded. For example, barrieroptions are rights to purchase an underlying financial product, such asa quantity of foreign currency, for a specified rate or price, but onlyif, for example, the underlying exchange rate crosses or does not crossone or more defined rates or “barriers.” For example, a dollar call/yenput on the dollar/yen exchange rate, expiring in three months withstrike price 110 and “knock-out” barrier of 105, entitles the holder topurchase a quantity of dollars at 110 yen per dollar, but only if theexchange rate did not fall below 105 at any point during the three monthduration of the option. Another example of a commonly traded exoticderivative, an Asian option, depends on the average value of theunderlying security over some time period. Thus, a class of exoticderivatives is commonly referred to as “path-dependent” derivatives,such as barrier and Asian options, since their values depend not only onthe value of the underlying financial product at a given date, but on ahistory of the value or state of the underlying financial product.

The properties and features of exotic derivatives are often so complexso as to present a significant source of “model risk” or the risk thatthe tools, or the assumptions upon which they are based, will lead tosignificant errors in pricing and hedging. Accordingly, derivativestraders and risk managers often employ sophisticated analytical tools totrade, hedge, and manage the risk of exotic derivatives.

One of the advantages of the systems and methods of the presentinvention is the ability to construct groups of DBAR contingent claimswith exotic features that are more manageable and transparent thantraditional exotic derivatives. For example, a trader might beinterested in the earliest time the yen/dollar exchange rate crosses 95over the next three months. A traditional barrier option, or portfolioof such exotic options, might suffice to approximate the source of riskof interest to this trader. A group of DBAR contingent claims, incontrast, can be constructed to isolate this risk and present relativelytransparent opportunities for hedging. A risk to be isolated is thedistribution of possible outcomes for what barrier derivatives tradersterm the “first passage time,” or, in this example, the first time thatthe yen/dollar exchange rate crosses 95 over the next three months.

The following illustration shows how such a group of DBAR contingentclaims can be constructed to address this risk. In this example, it isassumed that all traders in the group of claims agree that theunderlying exchange rate is lognormally distributed. This group ofclaims illustrates how traders would invest in states and thus expressopinions regarding whether and when the forward yen/dollar exchange ratewill cross a given barrier over the next 3 months:

-   -   Underlying Risk: Japanese/U.S. Dollar Yen Exchange Rate    -   Current Date Sep. 15, 1999    -   Expiration: Forward Rate First Passage Time, as defined, between        Sep. 16, 1999 to Dec. 16, 1999    -   Trading Start Date: Sep. 15, 1999    -   Trading End Date: Sep. 16, 1999    -   Barrier: 95    -   Spot JPY/USD: 104.68    -   Forward JPY/USD 103.268    -   Assumed (Illustrative) Market Volatility: 20% annualized    -   Aggregate Traded Amount: 10 million USD

TABLE 3.1.16-1 First Passage Time for Yen/Dollar Dec. 16, 1999 ForwardExchange Rate Time in Invested in Return Per Unit Year Fractions State(‘000) if State Occurs (0, .005] 229.7379 42.52786 (.005, .01] 848.902410.77992 (.01, .015] 813.8007 11.28802 (.015, .02] 663.2165 14.07803(.02, .025] 536.3282 17.6453 (.025 .03] 440.5172 21.70059 (.03, .035]368.4647 26.13964 (.035, .04] 313.3813 30.91 (.04, .045] 270.420735.97942 (.045, .05] 236.2651 41.32534 (.05, .075] 850.2595 10.76112(.075, .1] 540.0654 17.51627 (.1, .125] 381.3604 25.22191 (.125, .15]287.6032 33.77013 (.15, .175] 226.8385 43.08423 (.175, .2] 184.823853.10558 (.2, .225] 154.3511 63.78734 (.225, .25] 131.4217 75.09094 DidNot Hit Barrier 2522.242 2.964727

As with other examples, and in preferred embodiments, actual tradingwill likely generate traded amounts and therefore returns that departfrom the assumptions used to compute the illustrative returns for eachstate.

In addition to the straightforward multivariate events outlined above,demand-based markets or auctions can be used to create and trade digitaloptions (as described in Sections 6 and 7) on calculated underlyingevents (including the events described in this Section 3), similar tothose found in exotic derivatives. Many exotic derivatives are based onpath-dependent outcomes such as the average of an underlying event overtime, price thresholds, a multiple of the underlying, or some sort oftime constraint. An additional example follows:

-   -   Path Dependent: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, on an underlying event that is the        subject of a calculation. For example, digital options traded in        a demand-based market or auction could be based on an underlying        event defined as the average price of yen/dollar exchange rate        for the last quarter of 2001.

Example 3.1.17 Hedging Markets for Real Goods, Commodities and Services

Investment and capital budgeting choices faced by firms typicallyinvolve inherent economic risk (e.g., future demand for semiconductors),large capital investments (e.g., semiconductor fabrication capacity) andtiming (e.g., a decision to invest in a plant now, or defer for someperiod of time). Many economists who study such decisions underuncertainty have recognized that such choices involve what they term“real options.” This characterization indicates that the choice toinvest now or to defer an investment in goods or services or a plant,for example, in the face of changing uncertainty and information,frequently entails risks similar to those encountered by traders whohave invested in options which provide the opportunity to buy or sell anunderlying asset in the capital markets. Many economists and investorsrecognize the importance of real options in capital budgeting decisionsand of setting up markets to better manage their uncertainty and value.Natural resource and extractive industries, such as petroleumexploration and production, as well as industries requiring largecapital investments such as technology manufacturing, are prime examplesof industries where real options analysis is increasingly used andvalued.

Groups of DBAR contingent claims according to the present invention canbe used by firms within a given industry to better analyze capitalbudgeting decisions, including those involving real options. Forexample, a group of DBAR contingent claims can be established whichprovides hedging opportunities over the distribution of futuresemiconductor prices. Such a group of claims would allow producers ofsemiconductors to better hedge their capital budgeting decisions andprovide information as to the market's expectation of future prices overthe entire distribution of possible price outcomes. This informationabout the market's expectation of future prices could then also be usedin the real options context in order to better evaluate capitalbudgeting decisions. Similarly, computer manufacturers could use suchgroups of DBAR contingent claims to hedge against adverse semiconductorprice changes.

Information providing the basis for constructing an illustrative groupof DBAR contingent claims on semiconductor prices is as follows:

-   -   Underlying Event: Semiconductor Monthly Sales    -   Index: Semiconductor Industry Association Monthly Global Sales        Release    -   Current Date Sep. 15, 1999    -   Last Release Date: Sep. 2, 1999    -   Last Release Month: July, 1999    -   Last Release Value: 11.55 Billion, USD    -   Next Release Date: Approx. Oct. 1, 1999    -   Next Release Month: August 1999    -   Trading Start Date: Sep. 2, 1999    -   Trading End Date: Sep. 30, 1999

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in previous examples.

Groups of DBAR contingent claims according to the present invention canalso be used to hedge arbitrary sources of risk due to price discoveryprocesses. For example, firms involved in competitive bidding for goodsor services, whether by sealed bid or open bid markets or auctions, canhedge their investments and other capital expended in preparing the bidby investing in states of a group of DBAR contingent claims comprisingranges of mutually exclusive and collectively exhaustive market orauction bids. In this way, the group of DBAR contingent claim serves asa kind of “meta-auction,” and allows those who will be participating inthe market or auction to invest in the distribution of possible marketor auction outcomes, rather than simply waiting for the single outcomerepresenting the market or auction result. Market or auctionparticipants could thus hedge themselves against adverse market orauction developments and outcomes, and, importantly, have access to theentire probability distribution of bids (at least at one point in time)before submitting a bid into the real market or auction. Thus, a groupof DBAR claims could be used to provide market data over the entiredistribution of possible bids. Preferred embodiments of the presentinvention thus can help avoid the so-called Winner's Curse phenomenonknown to economists, whereby market or auction participants failrationally to take account of the information on the likely bids oftheir market or auction competitors.

Demand-based markets or auctions can be structured to offer a widevariety of products related to commodities such as fuels, chemicals,base metals, precious metals, agricultural products, etc. The followingexamples provide a further representative sampling:

-   -   Fuels: Demand-based markets or auctions can be based on measures        related to various fuel sources. For example, DBAR contingent        claims, including, e.g., digital options, can be based on an        underlying event defined as the price of natural gas in Btu's        delivered to the Henry Hub, Louisiana.    -   Chemicals: Demand-based markets or auctions can be based on        measures related to a variety of other chemicals. For example,        DBAR contingent claims, including, e.g., digital options, can be        based on an underlying event defined as the price of        polyethylene.    -   Base Metals Demand-based markets or auctions can be based on        measures related to various precious metals. For example, DBAR        contingent claims, including, e.g., digital options, can be        based on an underlying event defined as the price per gross ton        of #1 Heavy Melt Scrap Iron.    -   Precious Metals Demand-based markets or auctions can be based on        measures related to various precious metals. For example, DBAR        contingent claims, including, e.g., digital options, can be        based on an underlying event defined as the price per troy ounce        of Platinum delivered to an approved storage facility.    -   Agricultural Products: Demand-based markets or auctions can be        based on measures related to various agricultural products. For        example, DBAR contingent claims, including, e.g., digital        options, can be based on an underlying event defined as the        price per bushel of #2 yellow corn delivered at the Chicago        Switching District.

Example 3.1.18 DBAR Hedging

Another feature of the systems and methods of the present invention isthe relative ease with which traders can hedge risky exposures. In thefollowing example, it is assumed that a group of DBAR contingent claimshas two states (state 1 and state 2, or s₁ or s₂), and amounts T₁, andT₂ are invested in state 1 and state 2, respectively. The unit payout π₁for state 1 is therefore T₂/T₁ and for state 2 it is T₁/T₂. If a traderthen invests amount α₁ in state 1, and state 1 then occurs, the traderin this example would receive the following payouts, P, indexed by theappropriate state subscripts:

$P_{1} = {\alpha_{1}*\left( {\frac{T_{2}}{T_{1} + \alpha_{1}} + 1} \right)}$

If state 2 occurs the trader would receive

P₂=0

If, at some point during the trading period, the trader desires to hedgehis exposure, the investment in state 2 to do so is calculated asfollows:

$\alpha_{2} = \frac{\alpha_{1}*T_{2}}{T_{1}}$

This is found by equating the state payouts with the proposed hedgetrade, as follows:

$P_{1} = {{\alpha_{1}*\left( {\frac{T_{2} + \alpha_{2}}{T_{1} + \alpha_{1}} + 1} \right)} = {P_{2} = {\alpha_{2}*\left( {\frac{T_{1} + \alpha_{1}}{T_{2} + \alpha_{2}} + 1} \right)}}}$

Compared to the calculation required to hedge traditional derivatives,these expressions show that, in appropriate groups of DBAR contingentclaims of the present invention, calculating and implementing hedges canbe relatively straightforward.

The hedge ratio, α₂, just computed for a simple two state example can beadapted to a group of DBAR contingent claims which is defined over morethan two states. In a preferred embodiment of a group of DBAR contingentclaims, the existing investments in states to be hedged can bedistinguished from the states on which a future hedge investment is tobe made. The latter states can be called the “complement” states, sincethey comprise all the states that can occur other than those in whichinvestment by a trader has already been made, i.e., they arecomplementary to the invested states. A multi-state hedge in a preferredembodiment includes two steps: (1) determining the amount of the hedgeinvestment in the complement states, and (2) given the amount sodetermined, allocating the amount among the complement states. Theamount of the hedge investment in the complement states pursuant to thefirst step is calculated as:

$\alpha_{C} = \frac{\alpha_{H}*T_{C}}{T_{H}}$

where α_(C) is amount of the hedge investment in the complement states,α_(H) is the amount of the existing investment in the states to behedged, T_(c) is the existing amount invested in the complement states,and T_(H) is the amount invested the states to be hedged, exclusive ofα_(H). The second step involves allocating the hedge investment amongthe complement states, which can be done by allocating α_(c) among thecomplement states in proportion to the existing amounts already investedin each of those states.

An example of a four-state group of DBAR contingent claims according tothe present invention illustrates this two-step hedging process. Forpurposes of this example, the following assumptions are made: (i) thereare four states, numbered 1 through 4, respectively; (ii) $50, $80, $70and $40 is invested in each state, (iii) a trader has previously placeda multi-state investment in the amount of $10 (α_(H) as defined above)for states 1 and 2; and (iv) the allocation of this multi-stateinvestment in states 1 and 2 is $3.8462 and $6.15385, respectively. Theamounts invested in each state, excluding the trader's invested amounts,are therefore $46.1538, $73.84615, $70, and $40 for states 1 through 4,respectively. It is noted that the amount invested in the states to behedged, i.e., states 1 and 2, exclusive of the multi-state investment of$10, is the quantity T_(H) as defined above.

The first step in a preferred embodiment of the two-step hedging processis to compute the amount of the hedge investment to be made in thecomplement states. As derived above, the amount of the new hedgeinvestment is equal to the amount of the existing investment multipliedby the ratio of the amount invested in the complement states to theamount invested in the states to be hedged, excluding the trader'sexisting trades, i.e., $10*($70+$40)/($46.1538+$73.84615)=$9.16667. Thesecond step in this process is to allocate this amount between the twocomplement states, i.e., states 3 and 4.

Following the procedures discussed above for allocating multi-stateinvestments, the complement state allocation is accomplished byallocating the hedge investment amount—$9.16667 in this example—inproportion to the existing amount previously invested in the complementstates, i.e., $9.16667*$70/$110=$5.83333 for state 3 and$9.16667*$40/$110=$3.3333 for state 4. Thus, in this example, the tradernow has the following amounts invested in states 1 through 4: ($3.8462,$6.15385, $5.8333, $3.3333); the total amount invested in each of thefour states is $50, $80, $75.83333, and $43.3333); and the returns foreach of the four states, based on the total amount invested in each ofthe four states, would be, respectively, (3.98333, 2.1146, 2.2857, and4.75). In this example, if state 1 occurs the trader will receive apayout, including the amount invested in state 1, of3.98333*$3.8462+$3.8462=$19.1667 which is equal to the sum invested, sothe trader is fully hedged against the occurrence of state 1.Calculations for the other states yield the same results, so that thetrader in this example would be fully hedged irrespective of which stateoccurs.

As returns can be expected to change throughout the trading period, thetrader would correspondingly need to rebalance both the amount of hishedge investment for the complement states as well as the multi-stateallocation among the complement states. In a preferred embodiment, aDBAR contingent claim exchange can be responsible for reallocatingmulti-state trades via a suspense account, for example, so the tradercan assign the duty of reallocating the multi-state investment to theexchange. Similarly, the trader can also assign to an exchange theresponsibility of determining the amount of the hedge investment in thecomplement states especially as returns change as a result of trading.The calculation and allocation of this amount can be done by theexchange in a similar fashion to the way the exchange reallocatesmulti-state trades to constituent states as investment amounts change.

Example 3.1.19 Quasi-Continuous Trading

Preferred embodiments of the systems and methods of the presentinvention include a trading period during which returns adjust amongdefined states for a group of DBAR contingent claims, and a laterobservation period during which the outcome is ascertained for the eventon which the group of claims is based. In preferred embodiments, returnsare allocated to the occurrence of a state based on the finaldistribution of amounts invested over all the states at the end of thetrading period. Thus, in each embodiments a trader will not know hisreturns to a given state with certainty until the end of a given tradingperiod. The changes in returns or “price discovery” which occur duringthe trading period prior to “locking-in” the final returns may provideuseful information as to trader expectations regarding finalizedoutcomes, even though they are only indications as to what the finalreturns are going to be. Thus, in some preferred embodiments, a tradermay not be able to realize profits or losses during the trading period.The hedging illustration of Example 3.1.18, for instance, provides anexample of risk reduction but not of locking-in or realizing profit andloss.

In other preferred embodiments, a quasi-continuous market for trading ina group of DBAR contingent claims may be created. In preferredembodiments, a plurality of recurring trading periods may providetraders with nearly continuous opportunities to realize profit and loss.In one such embodiment, the end of one trading period is immediatelyfollowed by the opening of a new trading period, and the final investedamount and state returns for a prior trading period are “locked in” asthat period ends, and are allocated accordingly when the outcome of therelevant event is later known. As a new trading period begins on thegroup of DBAR contingent claims related to the same underlying event, anew distribution of invested amounts for states can emerge along with acorresponding new distribution of state returns. In such embodiments, asthe successive trading periods are made to open and close morefrequently, a quasi-continuous market can be obtained, enabling tradersto hedge and realize profit and loss as frequently as they currently doin the traditional markets.

An example illustrates how this feature of the present invention may beimplemented. The example illustrates the hedging of a European digitalcall option on the yen/dollar exchange rate (a traditional marketoption) over a two day period during which the underlying exchange ratechanges by one yen per dollar. In this example, two trading periods areassumed for the group of DBAR contingent claims

-   -   Traditional Option: European Digital Option    -   Payout of Option: Pays 100 million USD if exchange rate equals        or exceeds strike price at maturity or expiration    -   Underlying Index Yen/dollar exchange rate    -   Option Start Aug. 12, 1999    -   Option Expiration Aug. 15, 2000    -   Assumed Volatility: 20% annualized    -   Strike Price: 120    -   Notional: 100 million USD

In this example, two dates are analyzed, Aug. 12, 1999 and Aug. 13,1999:

TABLE 3.1.19-1 Change in Traditional Digital Call Option Value Over TwoDays Observation Date Aug. 12, 1999 Aug. 13, 1999 Spot Settlement DateAug. 16, 1999 Aug. 17, 1999 Spot Price for Settlement 115.55 116.55 DateForward Settlement Date Aug. 15, 2000 Aug. 15, 2000 Forward Price109.217107 110.1779 Option Premium 28.333% of Notional 29.8137% ofNotional

Table 3.1.19-1 shows how the digital call option struck at 120 could, asan example, change in value with an underlying change in the yen/dollarexchange rate. The second column shows that the option is worth 28.333%or $28.333 million on a $100 million notional on Aug. 12, 1999 when theunderlying exchange rate is 115.55. The third column shows that thevalue of the option, which pays $100 million should dollar yen equal orexceed 120 at the expiration date, increases to 29.8137% or $29.8137million per $100 million when the underlying exchange rate has increasedby 1 yen to 116.55. Thus, the traditional digital call option generatesa profit of $29.81377-$28.333=$1.48077 million.

This example shows how this profit also could be realized in trading ina group of DBAR contingent claims with two successive trading periods.It is also assumed for purposes of this example that there aresufficient amounts invested, or liquidity, in both states such that theparticular trader's investment does not materially affect the returns toeach state. This is a convenient but not necessary assumption thatallows the trader to take the returns to each state “as given” withoutconcern as to how his investment will affect the closing returns for agiven trading period. Using information from Table 3.1.19-1, thefollowing closing returns for each state can be derived:

Trading Period 1:

-   -   Current trading period end date: Aug. 12, 1999    -   Underlying Event Closing level of yen/dollar exchange rate for        Aug. 15, 2000 settlement, 4 pm EDT    -   Spot Price for Aug. 16, 1999 Settlement: 115.55

JPY/USD < 120 JPY/USD ≧ 120 State for Aug. 15, 2000 for Aug. 15, 2000Closing Returns 0.39533 2.5295

For purposes of this example, it is assumed that an illustrative traderhas $28.333 million invested in the state that the yen/dollar exchangerate equals or exceeds 120 for Aug. 15, 2000 settlement.

Trading Period 2:

-   -   Current trading period end date: Aug. 13, 1999    -   Underlying Event Closing level of dollar/yen exchange rate for        Aug. 15, 2000 settlement, 4 pm EDT    -   Spot Price for Aug. 17, 1999 Settlement: 116.55

JPY/USD < 120 JPY/USD ≧ 120 State for Aug. 15, 2000 for Aug. 15, 2000Closing State Returns .424773 2.3542

For purposes of this example, it is also assumed that the illustrativetrader has a $70.18755 million hedging investment in the state that theyen/dollar exchange rate is less than 120 for Aug. 15, 2000 settlement.It is noted that, for the second period, the closing returns are lowerfor the state that the exchange equals or exceeds 120. This is due tothe change represented in Table 3.1.19-1 reflecting an assumed change inthe underlying market, which would make that state more likely.

The trader now has an investment in each trading period and has lockedin a profit of $1.4807 million, as shown below:

JPY/USD < 120 for JPY/USD ≧ 120 State Aug. 15, 2000 for Aug. 15, 2000Profit and Loss $70.18755 * .424773 − $−70.18755 + 28.333 * (000.000)$28.333 = $1.48077 $2.5295 = $1.48077

The illustrative trader in this example has therefore been able tolock-in or realize the profit no matter which state finally occurs. Thisprofit is identical to the profit realized in the traditional digitaloption, illustrating that systems and methods of the present inventioncan be used to provide at least daily if not more frequent realizationof profits and losses, or that risks can be hedged in virtually realtime.

In preferred embodiments, a quasi-continuous time hedge can beaccomplished, in general, by the following hedge investment, assumingthe effect of the size of the hedge trade does not materially effect thereturns:

$H = {\alpha_{t}*\frac{1 + r_{t}}{1 + r_{t + 1}^{c}}}$

where

-   -   r_(t)=closing returns a state in which an investment was        originally made at time t    -   α_(t)=amount originally invested in the state at time t    -   r^(c) _(t+1)=closing returns at time t+1 to state or states        other than the state in which the original investment was made        (i.e., the so-called complement states which are all states        other than the state or states originally traded which are to be        hedged)    -   H=the amount of the hedge investment

If H is to be invested in more than one state, then a multi-stateallocation among the constituent states can be performed using themethods and procedures described above. This expression for H allowsinvestors in DBAR contingent claims to calculate the investment amountsfor hedging transactions. In the traditional markets, such calculationsare often complex and quite difficult.

Example 3.1.20 Value Units for Investments and Payouts

As previously discussed in this specification, the units of investmentsand payouts-used in embodiments of the present invention can be any unitof economic value recognized by investors, including, for example,currencies, commodities, number of shares, quantities of indices,amounts of swap transactions, or amounts of real estate. The investedamounts and payouts need not be in the same units and can comprise agroup or combination of such units, for example 25% gold, 25% barrels ofoil, and 50% Japanese Yen. The previous examples in this specificationhave generally used U.S. dollars as the value units for investments andpayouts.

This Example 3.1.20 illustrates a group of DBAR contingent claims for acommon stock in which the invested units and payouts are defined inquantities of shares. For this example, the terms and conditions ofExample 3.1.1 are generally used for the group of contingent claims onMSFT common stock, except for purposes of brevity, only three states arepresented in this Example 3.1.20: (0,83], (83, 88], and (88,∞]. Also inthis Example 3.1.20, invested amounts are in numbers of shares for eachstate and the exchange makes the conversion for the trader at the marketprice prevailing at the time of the investment. In this example, payoutsare made according to a canonical DRF in which a trader receives aquantity of shares equal to the number of shares invested in states thatdid not occur, in proportion to the ratio of number of shares the traderhas invested in the state that did occur, divided by the total number ofshares invested in that state. An indicative distribution of traderdemand in units of number of shares is shown below, assuming that thetotal traded amount is 100,000 shares:

Return Per Share if State Occurs Amount Traded in Number of Unit Returnsin Number of State Share Shares (0, 83] 17,803 4.617 (83, 88] 72,725.37504 (88, ∞] 9,472 9.5574

If, for instance, MSFT closes at 91 at expiration, then in this examplethe third state has occurred, and a trader who had previously invested10 shares in that state would receive a payout of 10*9.5574+10=105.574shares which includes the trader's original investment. Traders who hadpreviously invested in the other two states would lose all of theirshares upon application of the canonical DRF of this example.

An important feature of investing in value units other than units ofcurrency is that the magnitude of the observed outcome may well berelevant, as well as the state that occurs based on that outcome. Forexample, if the investments in this example were made in dollars, thetrader who has a dollar invested in state (88,∞] would not care, atleast in theory, whether the final price of MSFT at the close of theobservation period were 89 or 500. However, if the value units arenumbers of shares of stock, then the magnitude of the final outcome doesmatter, since the trader receives as a payout a number of shares whichcan be converted to more dollars at a higher outcome price of $91 pershare. For instance, for a payout of 105.574 shares, these shares areworth 105.574*$91=$9,607.23 at the outcome price. Had the outcome pricebeen $125, these shares would have been worth 105.574*125=$13,196.75.

A group of DBAR contingent claims using value units of commodity havinga price can therefore possess additional features compared to groups ofDBAR contingent claims that offer fixed payouts for a state, regardlessof the magnitude of the outcome within that state. These features mayprove useful in constructing groups of DBAR contingent claims which areable to readily provide risk and return profiles similar to thoseprovided by traditional derivatives. For example, the group of DBARcontingent claims described in this example could be of great interestto traders who transact in traditional derivatives known as“asset-or-nothing digital options” and “supershares options.”

Example 3.1.21 Replication of an Arbitrary Payout Distribution

An advantage of the systems and methods of the present invention isthat, in preferred embodiments, traders can generate an arbitrarydistribution of payouts across the distribution of defined states for agroup of DBAR contingent claims. The ability to generate a customizedpayout distribution may be important to traders, since they may desireto replicate contingent claims payouts that are commonly found intraditional markets, such as those corresponding to long positions instocks, short positions in bonds, short options positions in foreignexchange, and long option straddle positions, to cite just a fewexamples. In addition, preferred embodiments of the present inventionmay enable replicated distributions of payouts which can only begenerated with difficulty and expense in traditional markets, such asthe distribution of payouts for a long position in a stock that issubject to being “stopped out” by having a market-maker sell the stockwhen it reaches a certain price below the market price. Such stop-lossorders are notoriously difficult to execute in traditional markets, andtraders are frequently not guaranteed that the execution will occurexactly at the pre-specified price.

In preferred embodiments, and as discussed above, the generation andreplication of arbitrary payout distributions across a givendistribution of states for a group of DBAR contingent claims may beachieved through the use of multi-state investments. In suchembodiments, before making an investment, traders can specify a desiredpayout for each state or some of the states in a given distribution ofstates. These payouts form a distribution of desired payouts across thedistribution of states for the group of DBAR contingent claims. Inpreferred embodiments, the distribution of desired payouts may be storedby an exchange, which may also calculate, given an existing distributionof investments across the distribution of states, (1) the total amountrequired to be invested to achieve the desired payout distribution; (2)the states into which the investment is to allocated; and (3) how muchis to be invested in each state so that the desired payout distributioncan be achieved. In preferred embodiments, this multi-state investmentis entered into a suspense account maintained by the exchange, whichreallocates the investment among the states as the amounts investedchange across the distribution of states. In preferred embodiments, asdiscussed above, a final allocation is made at the end of the tradingperiod when returns are finalized.

The discussion in this specification of multi-state investments hasincluded examples in which it has been assumed that an illustrativetrader desires a payout which is the same no matter which state occursamong the constituent states of a multi-state investment. To achievethis result, in preferred embodiments the amount invested by the traderin the multi-state investment can be allocated to the constituent statein proportion to the amounts that have otherwise been invested in therespective constituent states. In preferred embodiments, theseinvestments are reallocated using the same procedure throughout thetrading period as the relative proportion of amounts invested in theconstituent states changes.

In other preferred embodiments, a trader may make a multi-stateinvestment in which the multi-state allocation is not intended togenerate the same payout irrespective of which state among theconstituent state occurs. Rather, in such embodiments, the multi-stateinvestment may be intended to generate a payout distribution whichmatches some other desired payout distribution of the trader across thedistribution of states, such as, for example, for certain digitalstrips, as discussed in Section 6. Thus, the systems and methods of thepresent invention do not require amounts invested in multi-stateinvestments to be allocated in proportion of the amounts otherwiseinvested in the constituent states of the multi-statement investment.

Notation previously developed in this specification is used to describea preferred embodiment of a method by which replication of an arbitrarydistribution of payouts can be achieved for a group of DBAR contingentclaims according to the present invention. The following additionalnotation, is also used:

-   -   A_(i,*) denotes the i-th row of the matrix A containing the        invested amounts by trader i for each of the n states of the        group of DBAR contingent claims        In preferred embodiments, the allocation of amounts invested in        all the states which achieves the desired payouts across the        distribution of states can be calculated using, for example, the        computer code listing in Table 1 (or functional equivalents        known to one of skill in the art), or, in the case where a        trader's multi-state investment is small relative to the total        investments already made in the group of DBAR contingent claims,        the following approximation:

A _(i,*) ^(T)=Π⁻¹ *P _(i,*) ^(T)

where the −1 superscript on the matrix Π denotes a matrix inverseoperation. Thus, in these embodiments, amounts to be invested to producean arbitrary distribution payouts can approximately be found bymultiplying (a) the inverse of a diagonal matrix with the unit payoutsfor each state on the diagonal (where the unit payouts are determinedfrom the amounts invested at any given time in the trading period) and(b) a vector containing the trader's desired payouts. The equation aboveshows that the amounts to be invested in order to produce a desiredpayout distribution are a function of the desired payout distributionitself (P_(i,*)) and the amounts otherwise invested across thedistribution of states (which are used to form the matrix Π, whichcontains the payouts per unit along its diagonals and zeroes along theoff-diagonals). Therefore, in preferred embodiments, the allocation ofthe amounts to be invested in each state will change if either thedesired payouts change or if the amounts otherwise invested across thedistribution change. As the amounts otherwise invested in various statescan be expected to change during the course of a trading period, inpreferred embodiments a suspense account is used to reallocate theinvested amounts, A_(i,*), in response to these changes, as describedpreviously. In preferred embodiments, at the end of the trading period afinal allocation is made using the amounts otherwise invested across thedistribution of states. The final allocation can typically be performedusing the iterative quadratic solution techniques embodied in thecomputer code listing in Table 1.

Example 3.1.21 illustrates a methodology for generating an arbitrarypayout distribution, using the event, termination criteria, the definedstates, trading period and other relevant information, as appropriate,from Example 3.1.1, and assuming that the desired multi-state investmentis small in relation to the total amount of investments already made. InExample 3.1.1 above, illustrative investments are shown across thedistribution of states representing possible closing prices for MSFTstock on the expiration date of Aug. 19, 1999. In that example, thedistribution of investment is illustrated for Aug. 18, 1999, one dayprior to expiration, and the price of MSFT on this date is given as 85.For purposes of this Example 3.1.21, it is assumed that a trader wouldlike to invest in a group of DBAR contingent claims according to thepresent invention in a way that approximately replicates the profits andlosses that would result from owning one share of MSFT (i.e., arelatively small amount) between the prices of 80 and 90. In otherwords, it is assumed that the trader would like to replicate atraditional long position in MSFT with the restrictions that a sellorder is to be executed when MSFT reaches 80 or 90. Thus, for example,if MSFT closes at 87 on Aug. 19, 1999 the trader would expect to have $2of profit from appropriate investments in a group of DBAR contingentclaims. Using the defined states identified in Example 3.1.1, thisprofit would be approximate since the states are defined to include arange of discrete possible closing prices.

In preferred embodiments, an investment in a state receives the samereturn regardless of the actual outcome within the state. It istherefore assumed for purposes of this Example 3.1.21 that a traderwould accept an appropriate replication of the traditional profit andloss from a traditional position, subject to only “discretization”error. For purposes of this Example 3.1.21, and in preferredembodiments, it is assumed that the profit and loss corresponding to anactual outcome within a state is determined with reference to the pricewhich falls exactly in between the upper and lower bounds of the stateas measured in units of probability, i.e., the “state average.” For thisExample 3.1.21, the following desired payouts can be calculated for eachof the states the amounts to be invested in each state and the resultinginvestment amounts to achieve those payouts:

TABLE 3.1.21-1 Investment Which Generates Desired States State Average($) Desired Payout ($) Payout ($) (0, 80] NA 80 0.837258 (80, 80.5]80.33673 80.33673 0.699493 (80.5, 81] 80.83349 80.83349 1.14091 (81,81.5] 81.33029 81.33029 1.755077 (81.5, 82] 81.82712 81.82712 2.549131(82, 82.5] 82.32401 82.32401 3.498683 (82.5, 83] 82.82094 82.820944.543112 (83, 83.5] 83.31792 83.31792 5.588056 (83.5, 84] 83.8149683.81496 6.512429 (84, 84.5] 84.31204 84.31204 7.206157 (84.5, 85]84.80918 84.80918 7.572248 (85, 85.5] 85.30638 85.30638 7.555924 (85.5,86] 85.80363 85.80363 7.18022 (86, 86.5] 86.30094 86.30094 6.493675(86.5, 87] 86.7983 86.7983 5.59628 (87, 87.5] 87.29572 87.29572 4.599353(87.5, 88] 87.7932 87.7932 3.611403 (88, 88.5] 88.29074 88.290742.706645 (88.5, 89] 88.78834 88.78834 1.939457 (89, 89.5] 89.2859989.28599 1.330046 (89.5, 90] 89.7837 89.7837 0.873212 (90, ∞] NA 901.2795The far right column of Table 3.1.21-1 is the result of the matrixcomputation described above. The payouts used to construct the matrix Πfor this Example 3.1.21 are one plus the returns shown in Example 3.1.1for each state.

Pertinently the systems and methods of the present invention may be usedto achieve almost any arbitrary payout or return profile, e.g., a longposition, a short position, an option “straddle”, etc., whilemaintaining limited liability and the other benefits of the inventiondescribed in this specification.

As discussed above, if many traders make multi-state investments, in apreferred embodiment an iterative procedure is used to allocate all ofthe multi-state investments to their respective constituent states.Computer code, as previously described and apparent to one of skill inthe art, can be implemented to allocate each multi-state investmentamong the constituent states depending upon the distribution of amountsotherwise invested and the trader's desired payout distribution.

Example 3.1.22 Emerging Market Currencies

Corporate and investment portfolio managers recognize the utility ofoptions to hedge exposures to foreign exchange movements. In the G7currencies, liquid spot and forward markets support an extremelyefficient options market. In contrast, many emerging market currencieslack the liquidity to support efficient, liquid spot and forward marketsbecause of their small economic base. Without ready access to a sourceof tradable underlying supply, pricing and risk control of options inemerging market currencies are difficult or impossible.

Governmental intervention and credit constraints further inhibittransaction flows in emerging market currencies. Certain governmentschoose to restrict the convertibility of their currency for a variety ofreasons, thus reducing access to liquidity at any price and effectivelypreventing option market-makers from gaining access to a tradableunderlying supply. Mismatches between sources of local liquidity andcreditworthy counterparties further restrict access to a tradableunderlying supply. Regional banks that service local customers haveaccess to indigenous liquidity but poor credit ratings whilemultinational commercial and investment banks with superior creditratings have limited access to liquidity. Because credit considerationsprevent external market participants from taking on significantexposures to local counterparties, transaction choices are limited.

The foreign exchange market has responded to this lack of liquidity bymaking use of non-deliverable forwards (NDFs) which, by definition, donot require an exchange of underlying currency. Although NDFs have metwith some success, their utility is still constrained by a lack ofliquidity. Moreover, the limited liquidity available to NDFs isgenerally insufficient to support an active options market.

Groups of DBAR contingent claims can be structured using the system andmethods of the present invention to support an active options market inemerging market currencies.

In addition to the general advantages of the demand-based tradingsystem, products on emerging market currencies will provide thefollowing new opportunities for trading and risk management:

-   (1) Credit enhancement. An investment bank can use demand-based    trading emerging market currency products to overcome existing    credit barriers. The ability of a demand-based market or auction to    process only buy orders, combined with the limited liability of    option payout profiles (vs. forward contracts), allows banks to    precisely define the limits of their counterparty credit exposure    and, hence, to trade with local market institutions, increasing    participation and liquidity.

Example 3.1.23 Central Bank Target Rates

Portfolio managers and market-makers formulate market views based inpart on their forecasts for future movements in central bank targetrates. When the Federal Reserve (Fed), European Central Bank (ECB) orBank of Japan (BOJ), for example, changes their target rate or whenmarket participants adjust their expectations about future rate moves,global equity and fixed income financial markets can react quickly anddramatically.

Market participants currently take views on central bank target rates bytrading 3-month interest rate futures, such as Eurodollar futures forthe Fed and Euribor futures for the ECB. Although these markets arequite liquid, significant risks impair trading in such contracts:futures contracts have a 3-month maturity while central bank targetrates change overnight; and models for credit spreads and term structureare required for futures pricing. Market participants additionallyexpress views on the target Fed funds rate by trading Fed funds futures,which are based on the overnight Fed funds rate. Although less riskythan Eurodollar futures, significant risks also impair trading in Fedfunds futures: the overnight Fed funds rate can differ, sometimessignificantly, from the target Fed funds rate due to overnight liquidityspikes and month-end effects; and, Fed funds futures frequently cannotaccommodate the full volumes that investment managers would like toexecute at a given market price.

Groups of DBAR contingent claims can be structured using the system andmethods of the present invention to develop an explicit mechanism bywhich market participants can express views regarding central banktarget rates. For example, demand-based markets or auctions can be basedon central bank policy parameters such as the Federal Reserve Target FedFunds Rate, the Bank of Japan Official Discount Rate, or the Bank ofEngland Base Rate. For example, the underlying event may be defined asthe Federal Reserve Target Fed Funds Rate as of Jun. 1, 2002. Becausedemand-based trading products settle using the target rate of interest,maturity and credit mismatches no longer pose market barriers.

In addition to the general advantages of the demand-based tradingsystem, products on central bank target rates may provide the followingnew advantages for trading and risk management:

-   (1) No basis risk. Since demand-based trading products settle using    the target rate of interest, there is no maturity mismatch and no    credit mismatch. Demand-based trading products for central bank    target rates have no basis risk.-   (2) An exact date match to central bank meetings. Demand-based    trading products can be structured to allow investors to take views    on specific meetings by matching the date of expiry of a contract    with the date of the central bank meeting.-   (3) A direct way to express views on intra-meeting moves.    Demand-based trading products allow special tailoring so that    portfolio managers can take a view on whether or not a central bank    will change its target rate intra-meeting.-   (4) Managing the event risk associated with a central bank meeting.    Almost all market participants have portfolios that are    significantly affected by shifts in target rates. Market    participants can use demand-based trading options on central bank    target rates to lower their portfolio's overall volatility.-   (5) Managing short-term funding costs. Banks and large corporations    often borrow short-term funds at a rate highly correlated with    central bank target rates, e.g., U.S. banks borrow at a rate that    closely follows target Fed funds. These institutions may better    manage their funding costs using demand-based trading products on    central bank rates.

Example 3.1.24 Weather

In recent years, market participants have expressed increasing interestin a market for derivative instruments related to weather as a means toinsure against adverse weather outcomes. Despite greater recognition ofthe role of weather in economic activity, the market for weatherderivatives has been relatively slow to develop. Market-makers intraditional over-the-counter markets often lack the means toredistribute their risk because of limited liquidity and lack of anunderlying instrument. The market for weather derivatives is furtherhampered by poor price discovery.

A group of DBAR contingent claims can be constructed using the methodsand systems of the present invention to provide market participants witha market price for the probability that a particular weather metric willbe above or below a given level. For example, participants in ademand-based market or auction on cooling degree days (CDDs) or onheating degree days (HDDs) in New York from Nov. 1, 2001 through Mar.31, 2002 may be able to see at a glance the market consensus price thatcumulative CDDs or HDDs will exceed certain levels. The eventobservation could be specified as taking place at a preset location suchas the Weather Bureau Army Navy Observation Station #14732.Alternatively, participants in a demand-based market or auction onwind-speed in Chicago may be able to see at a glance the marketconsensus price that cumulative wind-speeds will exceed certain levels.

Example 3.1.25 Financial Instruments

Demand-based markets or auctions can be structured to offer a widevariety of products on commonly offered financial instruments orstructured financial products related to fixed income securities,equities, foreign exchange, interest rates, and indices, and anyderivatives thereof. When the underlying economic event is a change (ordegree of change) in a financial instrument or product, the possibleoutcomes can include changes which are positive, negative or equal tozero when there is no change, and amounts of each positive and negativechange. The following examples provide a further representativesampling:

-   -   Equity Prices: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on prices for equity securities        listed on recognized exchanges throughout the world. For        example, DBAR contingent claims can be based on an underlying        event defined as the closing price each week of Juniper        Networks. The underlying event can also be defined using an        alternative measure, such as the volume weighted average price        during any day.    -   Fixed Income Security Prices: Demand-based markets or auctions        can be structured to trade DBAR contingent claims, including,        for example, digital options, based on a variety of fixed income        securities such as government T-bills, T-notes, and T-bonds,        commercial paper, CD's, zero coupon bonds, corporate, and        municipal bonds, and mortgage-backed securities. For example,        DBAR contingent claims can be based on an underlying event        defined as the closing price each week of Qwest Capital Funding        7¼% notes, due February of 2011. The underlying event can also        be defined using an alternative measure, such as the volume        weighted average price during any day. DBAR contingent claims on        government and municipal obligations can be traded in a similar        way.    -   Hybrid Security Prices: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on hybrid securities that        contain both fixed-income and equity features, such as        convertible bond prices. For example, DBAR contingent claims can        be based on an underlying event defined as the closing price        each week of Amazon.com 4¾% convertible bonds due February 2009.        The underlying event can also be defined using an alternative        measure, such as the volume weighted average price during any        day.    -   Interest Rates: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on interest rate measures such        as LIBOR and other money market rates, an index of AAA corporate        bond yields, or any of the fixed income securities listed above.        For example, DBAR contingent claims can be based on an        underlying event defined as the fixing price each week of        3-month LIBOR rates. Alternatively, the underlying event could        be defined as an average of an interest rate over a fixed length        of time, such as a week or month.    -   Foreign Exchange Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on foreign exchange rates. For        example, DBAR contingent claims can be based an underlying event        defined as the exchange rate of the Korean Won on any day.    -   Price & Return Indices: Demand-based markets or auctions can be        structured to trade DBAR contingent claims, including, for        example, digital options, based on a broad variety of financial        instrument price indices, including those for equities (e.g.,        S&P 500), interest rates, commodities, etc. For example, DBAR        contingent claims can be based on an underlying event defined as        the closing price each quarter of the S&P Technology index. The        underlying event can also be defined using an alternative        measure, such as the volume weighted average price during any        day. Alternatively, other index measurements can be used such as        return instead of price.    -   Swaps: Demand-based markets or auctions can be structured to        trade DBAR contingent claims, including, for example, digital        options, based on interest rate swaps and other swap based        transactions. In this example, discussed further in an        embodiment described in Section 9, digital options traded in a        demand-based market or auction are based on an underlying event        defined as the 10 year swap rate at which a fixed 10 year yield        is received against paying a floating 3 month LIBOR rate. The        rate may be determined using a common fixing convention.

Other derivatives on any security or other financial product orinstrument may be used as the underlying instrument for an event ofeconomic significance in a demand-based market or auction. For example,such derivatives can include futures, forwards, swaps, floating ratenotes and other structured financial products. Alternatively,derivatives strategies, securities (as well as other financial productsor instruments) and derivatives thereof can be converted into equivalentDBAR contingent claims or into replication sets of DBAR contingentclaims, such as digitals (for example, as in the embodiments discussedin Sections 9 and 10) and traded as a demand-enabled product alongsideDBAR contingent claims in the same demand-based market or auction.

3.2 DBAR Portfolios

It may be desirable to combine a number of groups of DBAR contingentclaims based on different events into a single portfolio. In this way,traders can invest amounts within the distribution of defined statescorresponding to a single event as well as across the distributions ofstates corresponding to all the groups of contingent claims in theportfolio. In preferred embodiments, the payouts to the amounts investedin this fashion can therefore be a function of a relative comparison ofall the outcome states in the respective groups of DBAR contingentclaims to each other. Such a comparison may be based upon the amountinvested in each outcome state in the distribution for each group ofcontingent claims as well as other qualities, parameters orcharacteristics of the outcome state (e.g., the magnitude of change foreach security underlying the respective groups of contingent claims). Inthis way, more complex and varied payout and return profiles can beachieved using the systems and methods of the present invention. Since apreferred embodiment of a demand reallocation function (DRF) can operateon a portfolio of DBAR contingent claims, such a portfolio is referredto as a DBAR Portfolio, or DBARP. A DBARP is a preferred embodiment ofDBAR contingent claims according to the present invention based on amulti-state, multi-event DRF.

In a preferred embodiment of a DBARP involving different events relatingto different financial products, a DRF is employed in which returns foreach contingent claim in the portfolio are determined by (i) the actualmagnitude of change for each underlying financial product and (ii) howmuch has been invested in each state in the distribution. A large amountinvested in a financial product, such as a common stock, on the longside will depress the returns to defined states on the long side of acorresponding group of DBAR contingent claims. Given the inverserelationship in preferred embodiments between amounts invested in andreturns from a particular state, one advantage to a DBAR portfolio isthat it is not prone to speculative bubbles. More specifically, inpreferred embodiments a massive influx of long side trading, forexample, will increase the returns to short side states, therebyincreasing returns and attracting investment in those states.

The following notation is used to explain further preferred embodimentsof DBARP:

-   -   μ_(i) is the actual magnitude of change for financial product i    -   W_(i) is the amount of successful investments in financial        product i    -   L_(i) is the amount of unsuccessful investments in financial        product i    -   f is the system transaction fee

${L\mspace{14mu} {is}\mspace{14mu} {the}\mspace{14mu} {aggregate}{\mspace{11mu} \;}{losses}} = {\sum\limits_{i}L_{i}}$${\gamma_{i}{\mspace{11mu} \;}{is}\mspace{14mu} {the}\mspace{14mu} {normalized}\mspace{14mu} {returns}{\mspace{11mu} \;}{for}\mspace{14mu} {successful}\mspace{14mu} {trades}} = \frac{\mu_{i}}{\sum\limits_{i}{\mu_{i}}}$

-   -   π^(p) _(i) is the payout per value unit invested in financial        product i for a successful investment    -   r^(p) _(i) is the return per unit invested in financial product        i for a successful investment

The payout principle of a preferred embodiment of a DBARP is to returnto a successful investment a portion of aggregate losses scaled by thenormalized return for the successful investment, and to return nothingto unsuccessful investments. Thus, in a preferred embodiment a largeactual return on a relatively lightly traded financial product willbenefit from being allocated a high proportion of the unsuccessfulinvestments.

$\pi_{i}^{p} = \frac{\gamma_{i}*L}{W_{i}}$${r_{i:=}^{p}\frac{\gamma_{i}*L}{W_{i}}} - 1$

As explained below, the correlations of returns across securities isimportant in preferred embodiments to determine payouts and returns in aDBARP.

An example illustrates the operation of a DBARP according to the presentinvention. For purposes of this example, it is assumed that a portfoliocontains two stocks, IBM and MSFT (Microsoft) and that the followinginformation applies (e.g., predetermined termination criteria):

-   -   Trading start date: Sep. 1, 1999    -   Expiration date: Oct. 1, 1999    -   Current trading period start date: Sep. 1, 1999    -   Current trading period end date: Sep. 5, 1999    -   Current date: Sep. 2, 1999    -   IBM start price: 129    -   MSFT start price: 96    -   Both IBM and MSFT Ex-dividends    -   No transaction fee

In this example, states can be defined so that traders can invest forIBM and MSFT to either depreciate or appreciate over the period. It isalso assumed that the distribution of amounts invested in the variousstates is the following at the close of trading for the current tradingperiod:

Financial Product Depreciate State Appreciate State MSFT $100 million$120 million IBM  $80 million  $65 millionThe amounts invested express greater probability assessments that MSFTwill likely appreciate over the period and IBM will likely depreciate.

For purposes of this example, it is further assumed that on theexpiration date of Oct. 1, 1999, the following actual outcomes forprices are observed:

-   -   MSFT: 106 (appreciated by 10.42%)    -   IBM 127 (depreciated by 1.55%)

In this example, there is $100+$65=$165 million to distribute from theunsuccessful investments to the successful investments, and, for thesuccessful investments, the relative performance of MSFT(10/42/(10.42+1.55)=0.871) is higher than for IBM °(1.55/10.42+1.55)=0.229). In a preferred embodiment, 87.1% of theavailable returns is allocated to the successful MSFT traders, with theremainder due the successful IBM traders, and with the following returnscomputed for each state:

${{\$ 120}\mspace{14mu} {million}\mspace{14mu} {of}\mspace{14mu} {successful}\mspace{14mu} {investment}\mspace{14mu} {produces}\mspace{14mu} a\mspace{14mu} {payout}\mspace{14mu} {of}\mspace{14mu} {.871}*{\$ 165}\mspace{14mu} {million}} = {{{{\$ 143}{.72}\mspace{14mu} {million}\mspace{14mu} {for}\mspace{14mu} a\mspace{14mu} {return}\mspace{14mu} {to}\mspace{14mu} {the}\mspace{14mu} {successful}\mspace{14mu} {traders}{\mspace{11mu} \;}{of}\mspace{14mu} \frac{{120M} + {143.72M}}{120M}} - 1} = {119.77\%}}$${{{\$ 80}\mspace{14mu} {million}\mspace{14mu} {in}\mspace{14mu} {successful}\mspace{14mu} {investment}\mspace{14mu} {produces}\mspace{14mu} a\mspace{14mu} {payout}\mspace{14mu} {of}\mspace{14mu} \left( {1 - {.871}} \right)*{\$ 165}\mspace{14mu} {million}} = {{\$ 21}{.285}\mspace{14mu} {million}}}\;,{{{{for}\mspace{14mu} a\mspace{14mu} {return}\mspace{14mu} {to}\mspace{14mu} {the}\mspace{14mu} {successful}\mspace{14mu} {traders}{\mspace{11mu} \;}{of}\mspace{14mu} \frac{{80M} + {21.285M}}{80M}} - 1} = {26.6\%}}$

The returns in this example and in preferred embodiments are a functionnot only of the amounts invested in each group of DBAR contingentclaims, but also the relative magnitude of the changes in prices for theunderlying financial products or in the values of the underlying eventsof economic performance. In this specific example, the MSFT tradersreceive higher returns since MSFT significantly outperformed IBM. Inother words, the MSFT longs were “more correct” than the IBM shorts.

The operation of a DBARP is further illustrated by assuming instead thatthe prices of both MSFT and IBM changed by the same magnitude, e.g.,MSFT went up 10%, and IBM went down 10%, but otherwise maintaining theassumptions for this example. In this scenario, $165 million of returnswould remain to distribute from the unsuccessful investments but theseare allocated equally to MSFT and IBM successful investments, or $82.5million to each. Under this scenario the returns are:

${\frac{{120M} + {82.5M}}{120M} - 1} = {68.75\%}$${\frac{{80M} + {82.5M}}{80M} - 1} = {103.125\%}$

The IBM returns in this scenario are 1.5 times the returns to the MFSTinvestments, since less was invested in the IBM group of DBAR contingentclaims than in the MSFT group.

This result confirms that preferred embodiments of the systems andmethods of the present invention provide incentives for traders to makelarge investments, i.e. promote liquidity, where it is needed in orderto have an aggregate amount invested sufficient to provide a fairindication of trader expectations.

The payouts in this example depend upon both the magnitude of change inthe underlying stocks as well as the correlations between such changes.A statistical estimate of these expected changes and correlations can bemade in order to compute expected returns and payouts during trading andat the close of each trading period. While making such an investment maybe somewhat more complicated that in a DBAR range derivative, asdiscussed above, it is still readily apparent to one of skill in the artfrom this specification or from practice of the invention.

The preceding example of a DBARP has been illustrated with eventscorresponding to closing prices of underlying securities. DBARPs of thepresent invention are not so limited and may be applied to any events ofeconomic significance, e.g., interest rates, economic statistics,commercial real estate rentals, etc. In addition, other types of DRFsfor use with DBARPs are apparent to one of ordinary skill in the art,based on this specification or practice of the present invention.

4. RISK CALCULATIONS

Another advantage of the groups of DBAR contingent claims according tothe present invention is the ability to provide transparent riskcalculations to traders, market risk managers, and other interestedparties. Such risks can include market risk and credit risk, which arediscussed below.

4.1 Market Risk

Market risk calculations are typically performed so that traders haveinformation regarding the probability distribution of profits and lossesapplicable to their portfolio of active trades. For all tradesassociated with a group of DBAR contingent claims, a trader might wantto know, for example, the dollar loss associated with the bottom fifthpercentile of profit and loss. The bottom fifth percentile correspondsto a loss amount which the trader knows, with a 95% statisticalconfidence, would not be exceeded. For the purposes of thisspecification, the loss amount associated with a given statisticalconfidence (e.g., 95%, 99%) for an individual investment is denoted thecapital-at-risk (“CAR”). In preferred embodiments of the presentinvention, a CAR can be computed not only for an individual investmentbut also for a plurality of investments related to for the same event orfor multiple events.

In the financial industry, there are three common methods that arecurrently employed to

compute CAR: (1) Value-at-Risk (“VAR”); (2) Monte Carlo Simulation(“MCS”); and (3) Historical Simulation (“HS”).

4.1.1 Capital-At-Risk Determinations Using Value-at-Risk Techniques

VAR is a method that commonly relies upon calculations of the standarddeviations and correlations of price changes for a group of trades.These standard deviations and correlations are typically computed fromhistorical data. The standard deviation data are typically used tocompute the CAR for each trade individually.

To illustrate the use of VAR with a group of DBAR contingent claims ofthe present invention, the following assumptions are made: (i) a traderhas made a traditional purchase of a stock, say $100 of IBM; (ii) usingpreviously computed standard deviation data, it is determined that theannual standard deviation for IBM is 30%; (iii) as is commonly the case,the price changes for IBM have a normal distribution; and (iv) thepercentile of loss to be used is the bottom fifth percentile. Fromstandard normal tables, the bottom fifth percentile of loss correspondsto approximately 1.645 standard deviations, so the CAR in thisexample—that is, loss for the IBM position that would not be exceededwith 95% statistical confidence—is 30%*1.645*$100, or $49.35. A similarcalculation, using similar assumptions, has been made for a $200position in GM, and the CAR computed for GM is $65.50. If, in thisexample, the computed correlation, ζ, between the prices of IBM and GMstock is 0.5, the CAR for the portfolio containing both the IBM and GMpositions may be expressed as:

$\begin{matrix}{{CAR} = \sqrt{\begin{matrix}{\left( {1.645\alpha_{IBM}\sigma_{IBM}} \right)^{2} + \left( {1.645\alpha_{GM}\sigma_{GM}} \right)^{2} +} \\{2{ϛ1}{.645}\alpha_{IBM}\sigma_{IBM}*1.645\alpha_{GM}\sigma_{GM}}\end{matrix}}} \\{= \sqrt{49.35^{2} + 65.50^{2} + {2*{.5}*49.35*65.5}}} \\{= 99.79}\end{matrix}$

where α is the investment in dollars, σ is the standard deviation, and ζis the correlation.

These computations are commonly represented in matrix form as:

-   -   C is the correlation matrix of the underlying events,    -   W is the vector containing the CAR for each active position in        the portfolio, and    -   W^(T) is the transpose of W.        In preferred embodiments, C is a y×y matrix, where y is the        number of active positions in the portfolio, and where the        elements of C are:    -   c_(i,j)=1 when i=j i.e., has l's on the diagonal, and otherwise    -   c_(i,j)=the correlation between the ith and jth events

$\begin{matrix}{{CAR} = \sqrt{w^{T}*C*w}} \\{= \sqrt{\begin{pmatrix}49.35 & 65.5\end{pmatrix}\begin{pmatrix}1 & {.5} \\{.5} & 1\end{pmatrix}\begin{pmatrix}49.35 \\65.5\end{pmatrix}}}\end{matrix}$

In preferred embodiments, several steps implement the VAR methodologyfor a group of DBAR contingent claims of the present invention. Thesteps are first listed, and details of each step are then provided. Thesteps are as follows:

(1) beginning with a distribution of defined states for a group of DBARcontingent claims, computing the standard deviation of returns in valueunits (e.g., dollars) for each investment in a given state;

(2) performing a matrix calculation using the standard deviation ofreturns for each state and the correlation matrix of returns for thestates within the same distribution of states, to obtain the standarddeviation of returns for all investments in a group of DBAR contingentclaims;

(3) adjusting the number resulting from the computation in step (2) foreach investment so that it corresponds to the desired percentile ofloss;

(4) arranging the numbers resulting from step (3) for each distinct DBARcontingent claim in the portfolio into a vector, w, having dimensionequal to the number of distinct DBAR contingent claims;

(5) creating a correlation matrix including the correlation of each pairof the underlying events for each respective DBAR contingent claim inthe portfolio; and

(6) calculating the square root of the product of w, the correlationmatrix created in step (5), and the transpose of w.

The result is CAR using the desired percentile of loss, for all thegroups of DBAR contingent claims in the portfolio.

In preferred embodiments, the VAR methodology of steps (1)-(6) above canbe applied to an arbitrary group of DBAR contingent claims as follows.For purposes of illustrating this methodology, it is assumed that allinvestments are made in DBAR range derivatives using a canonical DRF aspreviously described. Similar analyses apply to other forms of DRFs.

In step (1), the standard deviation of returns per unit of amountinvested for each state i for each group of DBAR contingent claim iscomputed as follows:

$\sigma_{i} = {\sqrt{\frac{T}{T_{i}} - 1} = {\sqrt{\frac{\left( {1 - q_{i}} \right)}{q_{i}}} = \sqrt{r_{i}}}}$

where σ_(i) is the standard deviation of returns per unit of amountinvested in each state i, T_(i) is the total amount invested in state i;T is the sum of all amounts invested across the distribution of states;q_(i) is the implied probability of the occurrence of state i derivedfrom T and T_(i); and r_(i) is the return per unit of investment instate i. In this preferred embodiment, this standard deviation is afunction of the amount invested in each state and total amount investedacross the distribution of states, and is also equal to the square rootof the unit return for the state. If α_(i) is the amount invested instate i, α_(i)*σ_(i) is the standard deviation in units of the amountinvested (e.g., dollars) for each state i.

Step (2) computes the standard deviation for all investments in a groupof DBAR contingent claims. This step (2) begins by calculating thecorrelation between each pair of states for every possible pair withinthe same distribution of states for a group of DBAR contingent claims.For a canonical DRF, these correlations may be computed as follows:

$\begin{matrix}{\rho_{i,j} = {- \frac{\sqrt{T_{i}*T_{j}}}{\sqrt{\left( {T - T_{i}} \right)*\left( {T - T_{j}} \right)}}}} \\{= {- \sqrt{\frac{q_{i}*q_{j}}{\left( {1 - q_{i}} \right)*\left( {1 - q_{j}} \right)}}}} \\{= \frac{- 1}{\sqrt{r_{i}*r_{j}}}} \\{= \frac{- 1}{\sigma_{i}*\sigma_{j}}}\end{matrix}$

where ρ_(i,j) is the correlation between state i and state j. Inpreferred embodiments, the returns to each state are negativelycorrelated since the occurrence of one state (a successful investment)precludes the occurrence of other states (unsuccessful investments). Ifthere are only two states in the distribution of states, thenT_(j)=T−T_(i) and the correlation ρ_(i,j) is −1, i.e., an investment instate i is successful and in state j is not, or vice versa, if i and jare the only two states. In preferred embodiments where there are morethan two states, the correlation falls in the range between 0 and −1(the correlation is exactly 0 if and only if one of the states hasimplied probability equal to one). In step (2) of the VAR methodology,the correlation coefficients ρ_(i,j) are put into a matrix C_(s) (thesubscript s indicating correlation among states for the same event)which contains a number of rows and columns equal to the number ofdefined states for the group of DBAR contingent claims. The correlationmatrix contains 1's along the diagonal, is symmetric, and the element atthe i-th row and j-th column of the matrix is equal to ρ_(i,j). Fromstep (1) above, a n×1 vector U is constructed having a dimension equalto the number of states n, in the group of DBAR contingent claims, witheach element of U being equal to α_(i)*σ_(i). The standard deviation,w_(k), of returns for all investments in states within the distributionof states defining the kth group of DBAR contingent claims can becalculated as follows:

w _(k)=√{square root over (U ^(T) *C _(s) *U)}

Step (3) involves adjusting the previously computed standard deviation,w_(k), for every group of DBAR contingent claims in a portfolio by anamount corresponding to a desired or acceptable percentile of loss. Forpurposes of illustration, it is assumed that investment returns have anormal distribution function; that a 95% statistical confidence forlosses is desirable; and that the standard deviations of returns foreach group of DBAR contingent claims, w_(k), can be multiplied by 1.645,i.e., the number of standard deviations in the standard normaldistribution corresponding to the bottom fifth percentile. A normaldistribution is used for illustrative purposes, and other types ofdistributions (e.g., the Student T distribution) can be used to computethe number of standard deviations corresponding to the any percentile ofinterest. As discussed above, the maximum amount that can be lost inpreferred embodiments of canonical DRF implementation of a group of DBARcontingent claims is the amount invested.

Accordingly, for this illustration the standard deviations w_(k) areadjusted to reflect the constraint that the most that can be lost is thesmaller of (a) the total amount invested and (b) the percentile loss ofinterest associated with the CAR calculation for the group of DBARcontingent claims, i.e.:

$w_{k} = {\min\left( {{1.645*w_{k}},{\sum\limits_{i = {1\mspace{14mu} \ldots \mspace{14mu} n}}\alpha_{i}}} \right)}$

In effect, this updates the standard deviation for each event bysubstituting for it a CAR value that reflects a multiple of the standarddeviation corresponding to an extreme loss percentile (e.g., bottomfifth) or the total invested amount, whichever is smaller.

Step (4) involves taking the adjusted w_(k), as developed in step (4)for each of m groups of DBAR contingent claims, and arranging them intoan y×1 dimensional column vector, w, each element of which containsw_(k), k=1 . . . y.

Step (5) involves the development of a symmetric correlation matrix,C_(e), which has a number of rows and columns equal to the number ofgroups of DBAR contingent claims, y, in which the trader has one or moreinvestments. Correlation matrix C_(e) can be estimated from historicaldata or may be available more directly, such as the correlation matrixamong foreign exchange rates, interest rates, equity indices,commodities, and other financial products available from JP Morgan'sRiskMetrics database. Other sources of the correlation information formatrix C_(e) are known to those of skill in the art. Along the diagonalsof the correlation matrix C_(e) are 1's, and the entry at the i-th rowand j-th column of the matrix contains the correlation between the i-thand j-th events which define the i-th and j-th DBAR contingent claim forall such possible pairs among the m active groups of DBAR contingentclaims in the portfolio.

In Step (6), the CAR for the entire portfolio of m groups of DBARcontingent claims is found by performing the following matrixcomputation, using each w_(k) from step (4) arrayed into vector w andits transpose w^(T):

CAR=√{square root over (w ^(T) *C _(e) *w)}

This CAR value for the portfolio of groups of DBAR contingent claims isan amount of loss that will not be exceeded with the associatedstatistical confidence used in Steps (1)-(6) above (e.g., in thisillustration, 95%).

Example 4.1.1-1 VAR-Based CAR Calculation

An example further illustrates the calculation of a VAR-based CAR for aportfolio containing two groups of DBAR range derivative contingentclaims (i.e., y=2) with a canonical DRF on two common stocks, IBM andGM. For this example, the following assumptions are made: (i) for eachof the two groups of DBAR contingent claims, the relevant underlyingevent upon which the states are defined is the respective closing priceof each stock one month forward; (ii) there are only three statesdefined for each event: “low”, “medium”, and “high,” corresponding toranges of possible closing prices on that date; (iii) the posted returnsfor IBM and GM respectively for the three respective states are, in U.S.dollars, (4, 0.6667, 4) and (2.333, 1.5, 2.333); (iv) the exchange feeis zero; (v) for the IBM group of contingent claims, the trader has onedollar invested in the state “low”, three dollars invested in the state“medium,” and two dollars invested in the state “high”; (vi) for the GMgroup of contingent claims, the trader has a single investment in theamount of one dollar in the state “medium”; (vii) the desired oracceptable percentile of loss in the fifth percentile, assuming a normaldistribution; and (viii) the estimated correlation of the price changesof IBM and GM is 0.5 across the distribution of states for each stock.

Steps (1)-(6), described above, are used to implement VAR in order tocompute CAR for this example. From Step (1), the standard deviations ofstate returns per unit of amount invested in each state for the IBM andGM groups of contingent claims are, respectively, (2, 0.8165, 2) and(1.5274, 1.225, 1.5274). In further accordance with Step (1) above, theamount invested in each state in the respective group of contingentclaims, α_(i); is multiplied by the previously calculated standarddeviation of state returns per investment, σ_(i), so that the standarddeviation of returns per state in dollars for each claim equals, for theIBM group: (2, 2.4495, 4) and, for the GM group, (0, 1.225, 0).

In accordance with Step (2) above, for each of the two groups of DBARcontingent claims in this example, a correlation matrix between any pairof states, C_(s), is constructed, as follows:

$C_{s}^{IBM} = \begin{matrix}1 & {- {.6124}} & {- {.25}} \\{- {.6124}} & 1 & {- {.6124}} \\{- {.25}} & {- {.6124}} & 1\end{matrix}$ $C_{s}^{GM} = \begin{matrix}1 & {- {.5345}} & {- {.4286}} \\{- {.5345}} & 1 & {- {.5345}} \\{- {.4286}} & {- {.5345}} & 1\end{matrix}$

where the left matrix is the correlation between each pair of statereturns for the IBM group of contingent claims and the right matrix isthe corresponding matrix for the GM group of contingent claims.

Also according to step (2) above, for each of the two groups ofcontingent claims, the standard deviation of returns per state indollars, α_(i)σ_(i) for each investment in this example can be arrangedin a vector with dimension equal to three (i.e., the number of states):

$U_{IBM} = \begin{matrix}2 \\2.4495 \\4\end{matrix}$ $U_{GM} = \begin{matrix}0 \\1.225 \\0\end{matrix}$

where the vector on the left contains the standard deviation in dollarsof returns per state for the IBM group of contingent claims, and thevector on the right contains the corresponding information for the GMgroup of contingent claims. Further in accordance with Step (2) above, amatrix calculation can be performed to compute the total standarddeviation for all investments in each of the two groups of contingentclaims, respectively:

w ₁=√{square root over (U _(IBM) ^(T) *C _(s) ^(IBM) *U _(IBM))}=2

w ₂=√{square root over (U _(GM) ^(T) *C _(s) ^(GM) *U _(GM))}=1.225

where the quantity on the left is the standard deviation for allinvestments in the distribution of the IBM group of contingent claims,and the quantity on the right is the corresponding standard deviationfor the GM group of contingent claims.

In accordance with step (3) above, w₁ and w₂ are adjusted by multiplyingeach by 1.645 (corresponding to a CAR loss percentile of the bottomfifth percentile assuming a normal distribution) and then taking thelower of (a) that resulting value and (b) the maximum amount that can belost, i.e., the amount invested in all states for each group ofcontingent claims:

w ₁=min(2*1.645,6)=3.29 w ₂=min(2*1.225,1)=1

where the left quantity is the adjusted standard deviation of returnsfor all investments across the distribution of the IBM group ofcontingent claims, and the right quantity is the corresponding amountinvested in the GM group of contingent claims. These two quantities, w₁and w₂, are the CAR values for the individual groups of DBAR contingentclaims respectively, corresponding to a statistical confidence of 95%.In other words, if the normal distribution assumptions that have beenmade with respect to the state returns are valid, then a trader, forexample, could be 95% confident that losses on the IBM groups ofcontingent claims would not exceed $3.29.

Proceeding now with Step (4) in the VAR process described above, thequantities w₁ and w₂ are placed into a vector which has a dimension oftwo, equal to the number of groups of DBAR contingent claims in theillustrative trader's portfolio:

$w = \begin{matrix}3.29 \\1\end{matrix}$

According to Step (5), a correlation matrix C_(e) with two rows and twocolumns, is either estimated from historical data or obtained from someother source (e.g., RiskMetrics), as known to one of skill in the art.Consistent with the assumption for this illustration that the estimatedcorrelation between the price changes of IBM and GM is 0.5, thecorrelation matrix for the underlying events is as follows:

$C_{e} = \begin{matrix}1 & {.5} \\{.5} & 1\end{matrix}$

Proceeding with Step (6), a matrix multiplication is performed by pre-and post-multiplying C_(e) by the transpose of w and by w, and takingthe square root of the resulting product:

CAR=√{square root over (w ^(T) *C _(e) *w)}=3.8877

This means that for the portfolio in this example, comprising the threeinvestments in the IBM group of contingent claims and the singleinvestment in the GM group of contingent claims, the trader can have a95% statistical confidence he will not have losses in excess of $3.89.

4.1.2 Capital-at-Risk Determinations Using Monte Carlo SimulationTechniques

Monte Carlo Simulation (“MCS”) is another methodology that is frequentlyused in the financial industry to compute CAR. MCS is frequently used tosimulate many representative scenarios for a given group of financialproducts, compute profits and losses for each representative scenario,and then analyze the resulting distribution of scenario profits andlosses. For example, the bottom fifth percentile of the distribution ofthe scenario profits and losses would correspond to a loss for which atrader could have a 95% confidence that it would not be exceeded. In apreferred embodiment, the MCS methodology can be adapted for thecomputation of CAR for a portfolio of DBAR contingent claims as follows.

Step (1) of the MCS methodology involves estimating the statisticaldistribution for the events underlying the DBAR contingent claims usingconventional econometric techniques, such as GARCH. If the portfoliobeing analyzed has more than one group of DBAR contingent claim, thenthe distribution estimated will be what is commonly known as amultivariate statistical distribution which describes the statisticalrelationship between and among the events in the portfolio. For example,if the events are underlying closing prices for stocks and stock pricechanges have a normal distribution, then the estimated statisticaldistribution would be a multivariate normal distribution containingparameters relevant for the expected price change for each stock, itsstandard deviation, and correlations between every pair of stocks in theportfolio. Multivariate statistical distribution is typically estimatedfrom historical time series data on the underlying events (e.g., historyof prices for stocks) using conventional econometric techniques.

Step (2) of the MCS methodology involves using the estimated statisticaldistribution of Step (1) in order to simulate the representativescenarios. Such simulations can be performed using simulation methodscontained in such reference works as Numerical Recipes in C or by usingsimulation software such as @Risk package available from Palisade, orusing other methods known to one of skill in the art. For each simulatedscenario, the DRF of each group of DBAR contingent claims in theportfolio determines the payouts and profits and losses on the portfoliocomputed.

Using the above two stock example involving GM and IBM used above todemonstrate VAR techniques for calculating CAR, a scenario simulated byMCS techniques might be “High” for IBM and “Low” for GM, in which casethe trader with the above positions would have a four dollar profit forthe IBM contingent claim and a one dollar loss for the GM contingentclaim, and a total profit of three dollars. In step (2), many suchscenarios are generated so that a resulting distribution of profit andloss is obtained. The resulting profits and losses can be arranged intoascending order so that, for example, percentiles corresponding to anygiven profit and loss number can be computed. A bottom fifth percentile,for example, would correspond to a loss for which the trader could be95% confident would not be exceeded, provided that enough scenarios havebeen generated to provide an adequate representative sample. This numbercould be used as the CAR value computed using MCS for a group of DBARcontingent claims. Additionally, statistics such as average profit orloss, standard deviation, skewness, kurtosis and other similarquantities can be computed from the generated profit and lossdistribution, as known by one of skill in the art.

4.1.3 Capital-at-Risk Determination Using Historical SimulationTechniques

Historical Simulation (“HS”) is another method used to compute. CARvalues. HS is comparable to that of MCS in that it relies upon the useof representative scenarios in order to compute a distribution of profitand loss for a portfolio. Rather than rely upon simulated scenarios froman estimated probability distribution, however, HS uses historical datafor the scenarios. In a preferred embodiment, HS can be adapted to applyto a portfolio of DBAR contingent claims as follows.

Step (1) involves obtaining, for each of the underlying eventscorresponding to each group of DBAR contingent claims, a historical timeseries of outcomes for the events. For example, if the events are stockclosing prices, time series of closing prices for each stock can beobtained from a historical database such as those available fromBloomberg, Reuters, or Datastream or other data sources known to someoneof skill in the art.

Step (2) involves using each observation in the historical data fromStep (1) to compute payouts using the DRF for each group of DBARcontingent claims in the portfolio. From the payouts for each group foreach historical observation, a portfolio profit and loss can becomputed. This results in a distribution of profits and lossescorresponding to the historical scenarios, i.e., the profit and lossthat would have been obtained had the trader held the portfoliothroughout the period covered by the historical data sample.

Step (3) involves arranging the values for profit and loss from thedistribution of profit and loss computed in Step (2) in ascending order.A profit and loss can therefore be computed corresponding to anypercentile in the distribution so arranged, so that, for example, a CARvalue corresponding to a statistical confidence of 95% can be computedby reference to the bottom fifth percentile.

4.2 Credit Risk

In preferred embodiments of the present invention, a trader may makeinvestments in a group of DBAR contingent claims using a margin loan. Inpreferred embodiments of the present invention implementing DBAR digitaloptions, an investor may make an investment with a profit and lossscenario comparable to a sale of a digital put or call option and thushave some loss if the option expires “in the money,” as discussed inSection 6, below. In preferred embodiments, credit risk may be measuredby estimating the amount of possible loss that other traders in thegroup of contingent claims could suffer owing to the inability of agiven trader to repay a margin loan or otherwise cover a loss exposure.For example, a trader may have invested $1 in a given state for a groupof DBAR contingent claims with $0.50 of margin. Assuming a canonical DRFfor this example, if the state later fails to occur, the DRF collects $1from the trader (ignoring interest) which would require repayment of themargin loan. As the trader may be unable to repay the margin loan at therequired time, the traders with successful trades may potentially not beable to receive the full amounts owing them under the DRF, and maytherefore receive payouts lower than those indicated by the finalizedreturns for a given trading period for the group of contingent claims.Alternatively, the risk of such possible losses due to credit risk maybe insured, with the cost of such insurance either borne by the exchangeor passed on to the traders. One advantage of the system and method ofthe present invention is that, in preferred embodiments, the amount ofcredit risk associated with a group of contingent claims can readily becalculated.

In preferred embodiments, the calculation of credit risk for a portfolioof groups of DBAR contingent claims involves computing acredit-capital-at-risk (“CCAR”) figure in a manner analogous to thecomputation of CAR for market risk, as described above.

The computation of CCAR involves the use of data related to the amountof margin used by each trader for each investment in each state for eachgroup of contingent claims in the portfolio, data related to theprobability of each trader defaulting on the margin loan (which cantypically be obtained from data made available by credit ratingagencies, such as Standard and Poors, and data related to thecorrelation of changes in credit ratings or default probabilities forevery pair of traders (which can be obtained, for example, from JPMorgan's CreditMetrics database).

In preferred embodiments, CCAR computations can be made with varyinglevels of accuracy and reliability. For example, a calculation of CCARthat is substantially accurate but could be improved with more data andcomputational effort may nevertheless be adequate, depending upon thegroup of contingent claims and the desires of traders for credit riskrelated information. The VAR methodology, for example, can be adapted tothe computation of CCAR for a group of DBAR contingent claims, althoughit is also possible to use MCS and HS related techniques for suchcomputations. The steps that can be used in a preferred embodiment tocompute CCAR using VAR-based, MCS-based, and HS-based methods aredescribed below.

4.2.1 CCAR Method for DBAR Contingent Claims Using the VAR-BasedMethodology

Step (i) of the VAR-based CCAR methodology involves obtaining, for eachtrader in a group of DBAR contingent claims, the amount of margin usedto make each trade or the amount of potential loss exposure from tradeswith profit and loss scenarios comparable to sales of options inconventional markets.

Step (ii) involves obtaining data related to the probability of defaultfor each trader who has invested in the groups of DBAR contingentclaims. Default probabilities can be obtained from credit ratingagencies, from the JP Morgan CreditMetrics database, or from othersources as known to one of skill in the art. In addition to defaultprobabilities, data related to the amount recoverable upon default canbe obtained. For example, an AA-rated trader with $1 in margin loans maybe able to repay $0.80 dollars in the event of default.

Step (iii) involves scaling the standard deviation of returns in unitsof the invested amounts. This scaling step is described in step (1) ofthe VAR methodology described above for estimating market risk. Thestandard deviation of each return, determined according to Step (1) ofthe VAR methodology previously described, is scaled by (a) thepercentage of margin [or loss exposure] for each investment; (b) theprobability of default for the trader; and (c) the percentage notrecoverable in the event of default.

Step (iv) of this VAR-based CCAR methodology involves taking from step(iii) the scaled values for each state for each investment andperforming the matrix calculation described in Step (2) above for theVAR methodology for estimating market risk, as described above. In otherwords, the standard deviations of returns in units of invested amountswhich have been scaled as described in Step (iii) of this CCARmethodology are weighted according to the correlation between eachpossible pair of states (matrix C_(s), as described above). Theresulting number is a credit-adjusted standard deviation of returns inunits of the invested amounts for each trader for each investment on theportfolio of groups of DBAR contingent claims. For a group of DBARcontingent claims, the standard deviations of returns that have beenscaled in this fashion are arranged into a vector whose dimension equalsthe number of traders.

Step (v) of this VAR-based CCAR methodology involves performing a matrixcomputation, similar to that performed in Step (5) of the VARmethodology for CAR described above. In this computation, the vector ofcredit-scaled standard deviations of returns from step (iv) are used topre- and post-multiply a correlation matrix with rows and columns equalto the number of traders, with 1's along the diagonal, and with theentry at row i and column j containing the statistical correlation ofchanges in credit ratings described above. The square root of theresulting matrix multiplication is an approximation of the standarddeviation of losses, due to default, for all the traders in a group ofDBAR contingent claims. This value can be scaled by a number of standarddeviations corresponding to a statistical confidence of thecredit-related loss not to be exceeded, as discussed above.

In a preferred embodiment, any given trader may be omitted from a CCARcalculation. The result is the CCAR facing the given trader due to thecredit risk posed by other traders who have invested in a group of DBARcontingent claims. This computation can be made for all groups of DBARcontingent claims in which a trader has a position, and the resultingnumber can be weighted by the correlation matrix for the underlyingevents, C_(e), as described in Step (5) for the VAR-based CARcalculation. The result corresponds to the risk of loss posed by thepossible defaults of other traders across all the states of all thegroups of DBAR contingent claims in a trader's portfolio.

4.2.2 CCAR Method for DBAR Contingent Claims Using the Monte CarloSimulation (MCS) Methodology

As described above, MCS methods are typically used to simulaterepresentative scenarios for a given group of financial products,compute profits and losses for each representative scenario, thenanalyze the resulting distribution of scenario profits and losses. Thescenarios are designed to be representative in that they are supposed tobe based, for instance, on statistical distributions which have beenestimated, typically using econometric time series techniques, to have agreat degree of relevance for the future behavior of the financialproducts. A preferred embodiment of MCS methods to estimate CCAR for aportfolio of DBAR contingent claims of the present invention, involvestwo steps, as described below.

Step (i) of the MCS methodology is to estimate a statisticaldistribution of the events of interest. In computing CCAR for a group ofDBAR contingent claims, the events of interest may be both the primaryevents underlying the groups of DBAR contingent claims, including eventsthat may be fitted to multivariate statistical distributions to computeCAR as described above, as well as the events related to the default ofthe other investors in the groups of DBAR contingent claims. Thus, in apreferred embodiment, the multivariate statistical distribution to beestimated relates to the market events (e.g., stock price changes,changes in interest rates) underlying the groups of DBAR contingentclaims being analyzed as well as the event that the investors in thosegroups of DBAR contingent claims, grouped by credit rating orclassification will be unable to repay margin loans for losinginvestments.

For example, a multivariate statistical distribution to be estimatedmight assume that changes in the market events and credit ratings orclassifications are jointly normally distributed. Estimating such adistribution would thus entail estimating, for example, the mean changesin the underlying market events (e.g., expected changes in interestrates until the expiration date), the mean changes in credit ratingsexpected until expiration, the standard deviation for each market eventand credit rating change, and a correlation matrix containing all of thepairwise correlations between every pair of events, including market andcredit event pairs. Thus, a preferred embodiment of MCS methodology asit applies to CCAR estimation for groups of DBAR contingent claims ofthe present invention typically requires some estimation as to thestatistical correlation between market events (e.g., the change in theprice of a stock issue) and credit events (e.g., whether an investorrated A− by Standard and Poors is more likely to default or bedowngraded if the price of a stock issue goes down rather than up).

It is sometimes difficult to estimate the statistical correlationsbetween market-related events such as changes in stock prices andinterest rates, on the one hand, and credit-related events such ascounterparty downgrades and defaults, on the other hand. Thesedifficulties can arise due to the relative infrequency of creditdowngrades and defaults. The infrequency of such credit-related eventsmay mean that statistical estimates used for MCS simulation can only besupported with low statistical confidence. In such cases, assumptionscan be employed regarding the statistical correlations between themarket and credit-related events. For example, it is not uncommon toemploy sensitivity analysis with regard to such correlations, i.e., toassume a given correlation between market and credit-related events andthen vary the assumption over the entire range of correlations from −1to 1 to determine the effect on the overall CCAR.

A preferred approach to estimating correlation between events is to usea source of data with regard to credit-related events that does nottypically suffer from a lack of statistical frequency. Two methods canbe used in this preferred approach. First, data can be obtained thatprovide greater statistical confidence with regard to credit-relatedevents. For example, expected default frequency data can be purchasedfrom such companies as KMV Corporation. These data supply probabilitiesof default for various parties that can be updated as frequently asdaily. Second, more frequently observed default probabilities can beestimated from market interest rates. For example, data providers suchas Bloomberg and Reuters typically provide information on the additionalyield investors require for investments in bonds of varying creditratings, e.g., AAA, AA, A, A−. Other methods are readily available toone skilled in the art to provide estimates regarding defaultprobabilities for various entities. Such estimates can be made asfrequently as daily so that it is possible to have greater statisticalconfidence in the parameters typically needed for MCS, such as thecorrelation between changes in default probabilities and changes instock prices, interest rates, and exchange rates.

The estimation of such correlations is illustrated assuming two groupsof DBAR contingent claims of interest, where one group is based upon theclosing price of IBM stock in three months, and the other group is basedupon the closing yield of the 30-year U.S. Treasury bond in threemonths. In this illustration, it is also assumed that the counterpartieswho have made investments on margin in each of the groups can be dividedinto five distinct credit rating classes. Data on the daily changes inthe price of IBM and the bond yield may be readily obtained from suchsources as Reuters or Bloomberg. Frequently changing data on theexpected default probability of investors can be obtained, for example,from KMV Corporation, or estimated from interest rate data as describedabove. As the default probability ranges between 0 and 1, a statisticaldistribution confined to this interval is chosen for purposes of thisillustration. For example, for purposes of this illustration, it can beassumed that the expected default probability of the investors follows alogistic distribution and that the joint distribution of changes in IBMstock and the 30-year bond yield follows a bivariate normaldistribution. The parameters for the logistic distribution and thebivariate normal distribution can be estimated using econometrictechniques known to one skilled in the art.

Step (ii) of a MCS technique, as it may be applied to estimating CCARfor groups of DBAR contingent claims, involves the use of themultivariate statistical distributions estimated in Step (i) above inorder to simulate the representative scenarios. As described above, suchsimulations can be performed using methods and software readilyavailable and known to those of skill in the art. For each simulatedscenario, the simulated default rate can be multiplied by the amount oflosses an investor faces based upon the simulated market changes and themargin, if any, the investor has used to make losing investments. Theproduct represents an estimated loss rate due to investor defaults. Manysuch scenarios can be generated so that a resulting distribution ofcredit-related expected losses can be obtained. The average value of thedistribution is the mean loss. The lowest value of the top fifthpercentile of the distribution, for example, would correspond to a lossfor which a given trader could be 95% confident would not be exceeded,provided that enough scenarios have been generated to provide astatistically meaningful sample. In preferred embodiments, the selectedvalue in the distribution, corresponding to a desired or adequateconfidence level, is used as the CCAR for the groups of DBAR contingentclaims being analyzed.

4.2.3 CCAR Method for DBAR Contingent Claims Using the HistoricalSimulation (“HS”) Methodology

As described above, Historical Simulation (HS) is comparable to MCS forestimating CCAR in that HS relies on representative scenarios in orderto compute a distribution of profit and loss for a portfolio of groupsof DBAR contingent claim investments. Rather than relying on simulatedscenarios from an estimated multivariate statistical distribution,however, HS uses historical data for the scenarios. In a preferredembodiment, HS methodology, for calculating CCAR for groups of DBARcontingent claims uses three steps, described below.

Step (i) involves obtaining the same data for the market-related eventsas described above in the context of CAR. In addition, to use HS toestimate CCAR, historical time series data are also used forcredit-related events such as downgrades and defaults. As such data aretypically rare, methods described above can be used to obtain morefrequently observed data related to credit events. For example, in apreferred embodiment, frequently-observed data on expected defaultprobabilities can be obtained from KMV Corporation. Other means forobtaining such data are known to those of skill in the art.

Step (ii) involves using each observation in the historical data fromthe previous step (i) to compute payouts using the DRF for each group ofDBAR contingent claims being analyzed. The amount of margin to be repaidfor the losing trades, or the loss exposure for investments with profitand loss scenarios comparable to digital option “sales,” can then bemultiplied by the expected default probability to use HS to estimateCCAR, so that an expected loss number can be obtained for each investorfor each group of contingent claims. These losses can be summed acrossthe investment by each trader so that, for each historical observationdata point, an expected loss amount due to default can be attributed toeach trader. The loss amounts can also be summed across all theinvestors so that a total expected loss amount can be obtained for allof the investors for each historical data point.

Step (iii) involves arranging, in ascending order, the values of lossamounts summed across the investors for each data point from theprevious step (ii). An expected loss amount due to credit-related eventscan therefore be computed corresponding to any percentile in thedistribution so arranged. For example, a CCAR value corresponding to a95% statistical confidence level can be computed by reference to 95^(th)percentile of the loss distribution.

5. LIQUIDITY AND PRICE/QUANTITY RELATIONSHIPS

In the trading of contingent claims, whether in traditional markets orusing groups of DBAR contingent claims of the present invention, it isfrequently useful to distinguish between the fundamental value of theclaim, on the one hand, as determined by market expectations,information, risk aversion and financial holdings of traders, and thedeviations from such value due to liquidity variations, on the otherhand. For example, the fair fundamental value in the traditional swapmarket for a five-year UK swap (i.e., swapping fixed interest forfloating rate payments based on UK LIBOR rates) might be 6.79% with a 2basis point bid/offer (i.e., 6.77% receive, 6.81% pay). A large traderwho takes the market's fundamental mid-market valuation of 6.79% ascorrect or fair might want to trade a swap for a large amount, such as750 million pounds. In light of likely liquidity available according tocurrent standards of the traditional market, the large amount of thetransaction could reduce the likely offered rate to 6.70%, which is afull 7 basis points lower than the average offer (which is probablyapplicable to offers of no more than 100 million pounds) and 9 basispoints away from the fair mid-market value.

The difference in value between a trader's position at the fair ormid-market value and the value at which the trade can actually becompleted, i.e. either the bid or offer, is usually called the liquiditycharge. For the illustrative five-year UK swap, a 1 basis pointliquidity charge is approximately equal to 0.04% of the amount traded,so that a liquidity charge of 9 basis points equals approximately 2.7million pounds. If no new information or other fundamental shocksintrude into or “hit” the market, this liquidity charge to the trader isalmost always a permanent transaction charge for liquidity—one that alsomust be borne when the trader decides to liquidate the large position.Additionally, there is no currently reliable way to predict, in thetraditional markets, how the relationship between price and quantity maydeviate from the posted bid and offers, which are usually applicableonly to limited or representative amounts. Price and quantityrelationships can be highly variable, therefore, due to liquidityvariations. Those relationships can also be non-linear. For instance, itmay cost more than twice as much, in terms of a bid/offer spread, totrade a second position that is only twice as large as a first position.

From the point of view of liquidity and transactions costs, groups ofDBAR contingent claims of the present invention offer advantagescompared to traditional markets. In preferred embodiments, therelationship between price (or returns) and quantity invested (i.e.,demanded) is determined mathematically by a DRF. In a preferredembodiment using a canonical DRF, the implied probability q_(i) for eachstate i increases, at a decreasing rate, with the amount invested inthat state:

$q_{i} = \frac{T_{i}}{T}$$\frac{\partial q_{i}}{\partial T_{i}} = \frac{T - T_{i}}{T^{2}}$$\frac{\partial^{2}q_{i}}{\partial T_{i}^{2}} = {{- 2}*\frac{T - T_{i}}{T^{3}}}$$\frac{\partial q_{i}}{\partial T_{j,{j \neq i}}} = {{- \frac{T_{i}}{T^{2}}} = {- \frac{q_{i}}{T}}}$

where T is the total amount invested across all the states of the groupof DBAR contingent claims and T_(i) is the amount invested in the statei. As a given the amount gets very large, the implied probability ofthat state asymptotically approaches one. The last expressionimmediately above shows that there is a transparent relationship,available to all traders, between implied probabilities and the amountinvested in states other than a given state i. The expression shows thatthis relationship is negative, i.e., as amounts invested in other statesincrease, the implied probability for the given state i decreases.Since, in preferred embodiments of the present invention, addinginvestments to states other than the given state is equivalent toselling the given state in the market, the expression for

$\frac{\partial q_{i}}{\partial T_{j,{j \neq i}}}$

above shows how, in a preferred embodiment, the implied probability forthe given state changes as a quantity for that state is up for sale,i.e., what the market's “bid” is for the quantity up for sale. Theexpression for

$\frac{\partial q_{i}}{\partial T_{i}}$

above shows, in a preferred embodiment, how the probability for thegiven state changes when a given quantity is demanded or desired to bepurchased, i.e., what the market's “offer” price is to purchasers of thedesired quantity.

In a preferred embodiment, for each set of quantities invested in thedefined states of a group of DBAR contingent claims, a set of bid andoffer curves is available as a function of the amount invested.

In the groups of DBAR contingent claims of the present invention, thereare no bids or offers per se. The mathematical relationships above areprovided to illustrate how the systems and methods of the presentinvention can, in the absence of actual bid/offer relationships, providegroups of DBAR contingent claims with some of the functionality ofbid/offer relationships.

Economists usually prefer to deal with demand and cross-demandelasticities, which are the percentage changes in prices due topercentage changes in quantity demanded for a given good (demandelasticity) or its substitute (cross-demand elasticity). In preferredembodiments of the systems and methods of the present invention, andusing the notation developed above,

${\frac{\Delta \; q_{i}}{q_{i}}/\frac{\Delta \; T_{i}}{T_{i}}} = {1 - q_{i}}$${\frac{\Delta \; q_{i}}{q_{i}}/\frac{\Delta \; T_{j}}{T_{j}}} = {- q_{j}}$

The first of the expressions immediately above shows that smallpercentage changes in the amount invested in state i have a decreasingpercentage effect on the implied probability for state i, as state ibecomes more likely (i.e., as q_(i) increases to 1). The secondexpression immediately above shows that a percentage change in theamount invested in a state j other than state i will decrease theimplied probability for state i in proportion to the implied probabilityfor the other state j.

In preferred embodiments, in order to effectively “sell” a state,traders need to invest or “buy” complement states, i.e., states otherthan the one they wish to “sell.” Thus, in a preferred embodimentinvolving a group of DBAR claims with two states, a “seller” of state 1will “buy” state 2, and vice versa. In order to “sell” state 1, state 2needs to be “bought” in proportion to the ratio of the amount investedin state 2 to the amount invested in state 1. In a state distributionwhich has more than two states, the “complement” for a given state to be“sold” are all of the other states for the group of DBAR contingentclaims. Thus, “selling” one state involves “buying” a multi-stateinvestment, as described above, for the complement states.

Viewed from this perspective, an implied offer is the resulting effecton implied probabilities from making a small investment in a particularstate. Also from this perspective, an implied bid is the effect onimplied probabilities from making a small multi-state investment incomplement states. For a given state in a preferred embodiment of agroup of DBAR contingent claims, the effect of an invested amount onimplied probabilities can be stated as follows:

${{Implied}\mspace{14mu} {``{Bid}"}} = {q_{i} - {\frac{\left( {1 - q_{i}} \right)}{T}*\Delta \; T_{i}}}$${{Implied}\mspace{14mu} {``{Offer}"}} = {q_{i} + {q_{i}*\left( {\frac{1}{T_{i}} - \frac{1}{T}} \right)*\Delta \; T_{i}}}$

where ΔT_(i) (considered here to be small enough for a first-orderapproximation) is the amount invested for the “bid” or “offer.” Theseexpressions for implied “bid” and implied “offer” can be used forapproximate computations. The expressions indicate how possibleliquidity effects within a group of DBAR contingent claims can be castin terms familiar in traditional markets. In the traditional markets,however, there is no ready way to compute such quantities for any givenmarket.

The full liquidity effect—or liquidity response function—between twostates in a group of DBAR contingent claims can be expressed asfunctions of the amounts invested in a given state, T_(i), and amountsinvested in the complement states, denoted T^(c) _(i), as follows:

${{Implied}\mspace{14mu} {``{Bid}"}\mspace{14mu} {Demand}\mspace{14mu} {Response}\mspace{14mu} {q_{i}^{B}\left( {\Delta \; T_{i}} \right)}} = \frac{T_{i}}{T_{i} + T_{i}^{c} + {\Delta \; T_{i}*\left( \frac{T_{i}^{c}}{T_{i} - {\Delta \; T_{i}}} \right)}}$${{Implied}\mspace{14mu} {``{Offer}"}\mspace{14mu} {Demand}\mspace{14mu} {Response}\mspace{14mu} {q_{i}^{O}\left( {\Delta \; T_{i}} \right)}} = \frac{T_{i} + {\Delta \; T_{i}}}{T_{i} + T_{i}^{c} + {\Delta \; T_{i}}}$

The implied “bid” demand response function shows the effect on theimplied state probability of an investment made to hedge an investmentof size ΔT_(i). The size of the hedge investment in the complementstates is proportional to the ratio of investments in the complementstates to the amount of investments in the state or states to be hedged,excluding the investment to be hedged (i.e., the third term in thedenominator). The implied “offer” demand response function above showsthe effect on the implied state probability from an incrementalinvestment of size ΔT_(i) in a particular defined state.

In preferred embodiments of systems and methods of the presentinvention, only the finalized returns for a given trading period areapplicable for computing payouts for a group of DBAR contingent claims.Thus, in preferred embodiments, unless the effect of a trade amount onreturns is permanent, i.e., persists through the end of a tradingperiod, a group of DBAR contingent claims imposes no permanent liquiditycharge, as the traditional markets typically do. Accordingly, inpreferred embodiments, traders can readily calculate the effect onreturns from investments in the DBAR contingent claims, and unless thesecalculated effects are permanent, they will not affect closing returnsand can, therefore, be ignored in appropriate circumstances. In otherwords, investing in a preferred embodiment of a group of DBAR contingentclaims does not impose a permanent liquidity charge on traders forexiting and entering the market, as the traditional markets typicallydo.

The effect of a large investment may, of course, move intra-tradingperiod returns in a group of DBAR contingent claims as indicated by theprevious calculations. In preferred embodiments, these effects couldwell be counteracted by subsequent investments that move the market backto fair value (in the absence of any change in the fundamental or fairvalue). In traditional markets, by contrast, there is usually a “tollbooth” effect in the sense that a toll or change is usually exactedevery time a trader enters and exits the market. This toll is largerwhen there is less “traffic” or liquidity and represents a permanentloss to the trader. By contrast, other than an exchange fee, inpreferred embodiments of groups of DBAR contingent claims, there is nosuch permanent liquidity tax or toll for market entry or exit.

Liquidity effects may be permanent from investments in a group of DBARcontingent claims if a trader is attempting to make a relatively verylarge investment near the end of a trading period, such that the marketmay not have sufficient time to adjust back to fair value. Thus, inpreferred embodiments, there should be an inherent incentive not to holdback large investments until the end of the trading period, therebyproviding incentives to make large investments earlier, which isbeneficial overall to liquidity and adjustment of returns. Nonetheless,a trader can readily calculate the effects on returns to a investmentwhich the trader thinks might be permanent (e.g., at the end of thetrading period), due to the effect on the market from a large investmentamount.

For example, in the two period hedging example (Example 3.1.19) above,it was assumed that the illustrated trader's investments had no materialeffect on the posted returns, in other words, that this trader was a“price taker.” The formula for the hedge trade H in the second period ofthat example above reflects this assumption. The following equivalentexpression for H takes account of the possibly permanent effect that alarge trade investment might have on the closing returns (because, forexample, the investment is made very close to the end of the tradingperiod):

$H = \frac{P_{t} - T_{t + 1} + \sqrt{T_{t + 1}^{2} - {2*T_{t + 1}*P_{t}} + P_{t}^{2} + {4*P_{t}*T_{t + 1}^{c}}}}{2}$

where

P_(t)=α_(t)*(1+r _(t))

in the notation used in Example 3.1.19, above, and T_(t+1) is the totalamount invested in period t+1 and T^(c) _(t+1) is the amount invested inthe complement state in period t+1. The expression for H is thequadratic solution which generates a desired payout, as described abovebut using the present notation. For example, if $1 billion is the totalamount, T, invested in trading period 2, then, according to the aboveexpressions, the hedge trade investment assuming a permanent effect onreturns is $70.435 million compared to $70.18755 million in Example3.1.19. The amount of profit and loss locked-in due to the new hedge is$1.232 million, compared to $1.48077 in Example 3.1.19. The differencerepresents the liquidity effect, which even in the example where theinvested notional is 10% of the total amount invested, is quitereasonable in a market for groups of DBAR contingent claims. There is noready way to estimate or calculate such liquidity effects in traditionalmarkets.

6. DBAR DIGITAL OPTIONS EXCHANGE

In a preferred embodiment, the DBAR methods and systems of the presentinvention may be used to implement financial products known as digitaloptions and to facilitate an exchange in such products. A digital option(sometimes also known as a binary option) is a derivative security whichpays a fixed amount should specified conditions be met (such as theprice of a stock exceeding a given level or “strike” price) at theexpiration date. If the specified conditions are met, a digital optionis often characterized as finishing “in the money.” A digital calloption, for example, would pay a fixed amount of currency, say onedollar, should the value of the underlying security, index, or variableupon which the option is based expire at or above the strike price ofthe call option. Similarly, a digital put option would pay a fixedamount of currency should the value of the underlying security, index orvariable be at or below the strike price of the put option. A spread ofeither digital call or put options would pay a fixed amount should theunderlying value expire at or between the strike prices. A strip ofdigital options would pay out fixed ratios should the underlying expirebetween two sets of strike prices. Graphically, digital calls, puts,spreads, and strips can have simple representations:

TABLE 6.0.1 Digital Call

TABLE 6.0.2 Digital Put

TABLE 6.0.3 Digital Spread

TABLE 6.0.4 Digital Strip

As depicted in Tables 6.0.1, 6.0.2, 6.0.3, and 6.0.4, the strike pricesfor the respective options are marked using familiar options notationwhere the subscript “c” indicates a call, the subscript “p” indicates aput, the subscript “s” indicates “spread,” and the superscripts “1” and“u” indicate lower and upper strikes, respectively.

A difference between digital options, which are frequently transacted inthe OTC foreign currency options markets, and traditional options suchas the equity options, which trade on the Chicago Board Options Exchange(“CBOE”), is that digital options have payouts which do not vary withthe extent to which the underlying asset, index, or variable(“underlying”) finishes in or out of the money. For example, a digitalcall option at a strike price for the underlying stock at 50 would paythe same amount if, at the fulfillment of all of the terminationcriteria, the underlying stock price was 51, 60, 75 or any other valueat or above 50. In this sense, digital options represent the academicfoundations of options theory, since traditional equity options could intheory be replicated from a portfolio of digital spread options whosestrike prices are set to provide vanishingly small spreads. (In fact, a“butterfly spread” of the traditional options yields a digital optionspread as the strike prices of the traditional options are allowed toconverge.) As can be seen from Tables 6.0.1, 6.0.2, 6.0.3, and 6.0.4,digital options can be constructed from digital option spreads.

The methods and systems of the present invention can be used to create aderivatives market for digital options spreads. In other words, eachinvestment in a state of a mutually exclusive and collectivelyexhaustive set of states of a group of DBAR contingent claims can beconsidered to correspond to either a digital call spread or a digitalput spread. Since digital spreads can readily and accurately be used toreplicate digital options, and since digital options are known, tradedand processed in the existing markets, DBAR methods can therefore berepresented effectively as a market for digital options—that is, a DBARdigital options market.

6.1 Representation of Digital Options as DBAR Contingent Claims

One advantage of the digital options representation of DBAR contingentclaims is that the trader interface of a DBAR digital options exchange(a “DBAR DOE”) can be presented in a format familiar to traders, eventhough the underlying DBAR market structure is quite novel and differentfrom traditional securities and derivatives markets. For example, themain trader interface for a DBAR digital options exchange, in apreferred embodiment, could have the following features:

TABLE 6.1.1 MSFT Digital Options CALLS PUTS IND IND IND IND IND INDSTRIKE BID OFFER PAYOUT BID OFFER PAYOUT 30 0.9388 0.9407 1.0641 0.05930.0612 16.5999 40 0.7230 0.7244 1.3818 0.2756 0.2770 3.6190 50 0.43990.4408 2.2708 0.5592 0.5601 1.7869 60 0.2241 0.2245 4.4582 0.7755 0.77591.2892 70 0.1017 0.1019 9.8268 0.8981 0.8983 1.1133 80 0.0430 0.043123.2456 0.9569 0.9570 1.0450The illustrative interface of Table 6.1.1 contains hypothetical marketinformation on DBAR digital options on Microsoft stock (“MSFT”) for agiven expiration date. For example, an investor who desires a payout ifMSFT stock closes higher than 50 at the expiration or observation datewill need to “pay the offer” of $0.4408 per dollar of payout. Such anoffer is “indicative” (abbreviated “IND”) since the underlying DBARdistribution—that is, the implied probability that a state or set ofstates will occur—may change during the trading period. In a preferredembodiment, the bid/offer spreads presented in Table 6.1.1 are presentedin the following manner. The “offer” side in the market reflects theimplied probability that underlying value of the stock (in this exampleMSFT) will finish “in the money.” The “bid” side in the market is the“price” at which a claim can be “sold” including the transaction fee.(In this context, the term “sold” reflects the use of the systems andmethods of the present invention to implement investment profit and lossscenarios comparable to “sales” of digital options, discussed in detailbelow.) The amount in each “offer” cell is greater than the amount inthe corresponding “bid” cell. The bid/offer quotations for these digitaloption representations of DBAR contingent claims are presented aspercentages of (or implied probabilities for) a one dollar indicativepayout.

The illustrative quotations in Table 6.1.1 can be derived as follows.First the payout for a given investment is computed assuming a 10 basispoint transaction fee. This payout is equal to the sum of allinvestments less 10 basis points, divided by the sum of the investmentsover the range of states corresponding to the digital option. Taking theinverse of this quantity gives the offer side of the market in “price”terms. Performing the same calculation but this time adding 10 basispoints to the total investment gives the bid side of the market.

In another preferred embodiment, transaction fees are assessed as apercentage of payouts, rather than as a function of invested amounts.Thus, the offer (bid) side of the market for a given digital optioncould be, for example, (a) the amount invested over the range of statescomprising the digital option, (b) plus (minus) the fee (e.g., 10 basispoints) multiplied by the total invested for all of the defined states,(c) divided by the total invested for all of the defined states. Anadvantage of computing fees based upon the payout is that the bid/offerspreads as a percentage of “price” would be different depending upon thestrike price of the underlying, with strikes that are less likely to be“in the money” having a higher percentage fee. Other embodiments inwhich the exchange or transaction fees, for example, depend on the timeof trade to provide incentives for traders to trade early or to tradecertain strikes, or otherwise reflect liquidity conditions in thecontract, are apparent to those of skill in the art.

As explained in detail below, in preferred embodiments of the systemsand methods of the present invention, traders or investors can buy and“sell” DBAR contingent claims that are represented and behave likedigital option puts, calls, spreads, and strips using conditional or“limit” orders. In addition, these digital options can be processedusing existing technological infrastructure in place at currentfinancial institutions. For example, Sungard, Inc., has a largesubscriber base to many off-the-shelf programs which are capable ofvaluing, measuring the risk, clearing, and settling digital options.Furthermore, some of the newer middleware protocols such as FINXML (seewww.finxml.org) apparently are able to handle digital options and otherswill probably follow shortly (e.g., FPML). In addition, the transactioncosts of a digital options exchange using the methods and systems of thepresent invention can be represented in a manner consistent with theconventional markets, i.e., in terms of bid/offer spreads.

6.2 Construction of Digital Options Using DBAR Methods and Systems

The methods of multistate trading of DBAR contingent claims previouslydisclosed can be used to implement investment in a group of DBARcontingent claims that behave like a digital option. In particular, andin a preferred embodiment, this can be accomplished by allocating aninvestment, using the multistate methods previously disclosed, in such amanner that the same payout is received from the investment should theoption expire “in-the-money”, e.g., above the strike price of theunderlying for a call option and below the strike price of theunderlying for a put. In a preferred embodiment, the multistate methodsused to allocate the investment need not be made apparent to traders. Insuch an embodiment, the DBAR methods and systems of the presentinvention could effectively operate “behind the scenes” to improve thequality of the market without materially changing interfaces and tradingscreens commonly used by traders. This may be illustrated by consideringthe DBAR construction of the MSFT Digital Options market activity asrepresented to the user in Table 6.1.1. For purposes of thisillustration, it is assumed that the market “prices” or impliedprobabilities for the digital put and call options as displayed in Table6.1.1 result from $100 million in investments. The DBAR states andallocated investments that construct these “prices” are then:

TABLE 6.2.1 States State Prob State Investments (0, 30] 0.0602387$6,023,869.94 (30, 40] 0.2160676 $21,606,756.78 (40, 50] 0.2833203$28,332,029.61 (50, 60] 0.2160677 $21,606,766.30 (60, 70] 0.1225432$12,254,324.67 (70, 80] 0.0587436 $5,874,363.31 (80, ∞] 0.0430189$4,301,889.39In Table 6.2.1, the notation (x, y] is used to indicate a single statepart of a set of mutually exclusive and collectively exhaustive stateswhich excludes x and includes y on the interval.

(For purposes of this specification a convention is adopted for puts,calls, and spreads which is consistent with the internal representationof the states. For example, a put and a call both struck at 50 cannotboth be paid out if the underlying asset, index or variable expiresexactly at 50. To address this issue, the following convention could beadopted: calls exclude the strike price, puts include the strike price,and spreads exclude the lower and include the upper strike price. Thisconvention, for example, would be consistent with internal states thatare exclusive on the lower boundary and inclusive on the upper boundary.Another preferred convention would have calls including the strike priceand puts excluding the strike price, so that the representation of thestates would be inclusive on the lower boundary and exclusive on theupper. In any event, related conventions exist in traditional markets.For example, consider the situation of a traditional foreign exchangeoptions dealer who sells an “at the money” digital and an “at the money”put, with strike price of 100. Each is equally likely to expire “in themoney,” so for every $1.00 in payout, the dealer should collect $0.50.If the dealer has sold a $1.00 digital call and put, and has thereforecollected a total of $1.00 in premium, then if the underlying expiresexactly at 100, a discontinuous payout of $2.00 is owed. Hence, in apreferred embodiment of the present invention, conventions such as thosedescribed above or similar methods may be adopted to avoid suchdiscontinuities.)

A digital call or put may be constructed with DBAR methods of thepresent invention by using the multistate allocation algorithmspreviously disclosed. In a preferred embodiment, the construction of adigital option involves allocating the amount to be invested across theconstituent states over which the digital option is “in-the-money”(e.g., above the strike for a call, below the strike for a put) in amanner such that the same payout is obtained regardless of which stateoccurs among the “in the money” constituent states. This is accomplishedby allocating the amount invested in the digital option in proportion tothe then-existing investments over the range of constituent states forwhich the option is “in the money.” For example, for an additional$1,000,000 investment a digital call struck at 50 from the investmentsillustrated in Table 6.2.1, the construction of the trade usingmultistate allocation methods is:

TABLE 6.2.2 Internal States $ 1,000,000.00 (0, 30] (30, 40] (40, 50](50, 60] $490,646.45 (60, 70] $278,271.20 (70, 80] $133,395.04 (80, ∞]$97,687.30As other traders subsequently make investments, the distribution ofinvestments across the states comprising the digital option may change,and may therefore require that the multistate investments be reallocatedso that, for each digital option, the payout is the same for any of itsconstituent “in the money” states, regardless of which of theseconstituent states occurs after the fulfillment of all of thetermination criteria, and is zero for any of the other states. When theinvestments have been allocated or reallocated so that this payoutscenario occurs, the group of investments or contract is said to be inequilibrium. A further detailed description of the allocation methodswhich can be used to achieve this equilibrium is provided in connectionwith the description of FIGS. 13-14.

6.3 Digital Option Spreads

In a preferred embodiment, a digital option spread trade may be offeredto investors which simultaneously execute a buy and a “sell” (in thesynthetic or replicated sense of the term, as described below) of adigital call or put option. An investment in such a spread would havethe same payout should the underlying outcome expire at any valuebetween the lower and upper strike prices in the spread. If the spreadcovers one state, then the investment is comparable to an investment ina DBAR contingent claim for that one state. If the spread covers morethan one constituent state, in a preferred embodiment the investment isallocated using the multistate investment method previously described sothat, regardless of which state occurs among the states included in thespread trade, the investor receives the same payout.

6.4 Digital Option Strips

Traders in the derivatives markets commonly trade related groups offutures or options contracts in desired ratios in order to accomplishsome desired purpose. For example, it is not uncommon for traders ofLIBOR based interest rate futures on the Chicago Mercantile Exchange(“CME”) to execute simultaneously a group of futures with differentexpiration dates covering a number of years. Such a group, which iscommonly termed a “strip,” is typically traded to hedge another positionwhich can be effectively approximated with a strip whose constituentcontracts are executed in target relative ratios. For example, a stripof LIBOR-based interest rate futures may be used to approximate the riskinherent of an interest rate swap of the same maturity as the latestcontract expiration date in the strip.

In a preferred embodiment, the DBAR methods of the present invention canbe used to allow traders to construct strips of digital options anddigital option spreads whose relative payout ratios, should each optionexpire in the money, are equal to the ratios specified by the trader.For example, a trader may desire to invest in a strip consisting of the50, 60, 70, and 80 digital call options on MSFT, as illustrated in Table6.1.1. Furthermore, and again as an illustrative example, the trader maydesire that the payout ratios, should each option expire in the money,be in the following relative ratio: 1:2:3:4. Thus, should the underlyingprice of MSFT at the expiration date (when the event outcome isobserved) be equal to 65, both the 50 and 60 strike digital options arein the money. Since the trader desires that the 60 strike digital calloption pay out twice as much as the 50 strike digital call option, amultistate allocation algorithm, as previously disclosed and describedin detail, can be used dynamically to reallocate the trader'sinvestments across the states over which these options are in the money(50 and above, and 60 and above, respectively) in such a way as togenerate final payouts which conform to the indicated ratio of 1:2. Aspreviously disclosed, the multistate allocation steps may be performedeach time new investments are added during the trading period, and afinal multistate allocation may be performed after the trading periodhas expired.

6.5 Multistate Allocation Algorithm for Replicating “Sell” Trades

In a preferred embodiment of a digital options exchange using DBARmethods and systems of the present invention, traders are able to makeinvestments in DBAR contingent claims which correspond to purchases ofdigital options. Since DBAR methods are inherently demand-based—i.e., aDBAR exchange or market functions without traditional sellers—anadvantage of the multistate allocation methods of the present inventionis the ability to generate scenarios of profits and losses (“P&L”)comparable to the P&L scenarios obtained from selling digital options,spreads, and strips in traditional, non-DBAR markets without traditionalsellers or order-matching.

In traditional markets, the act of selling a digital option, spread, orstrip means that the investor (in the case of a sale, a seller) receivesthe cost of the option, or premium, if the option expires worthless orout of the money. Thus, if the option expires out of the money, theinvestor/seller's profit is the premium. Should the option expire in themoney, however, the investor/seller incurs a net liability equal to thedigital option payout less the premium received. In this situation, theinvestor/seller's net loss is the payout less the premium received forselling the option, or the notional payout less the premium. Selling anoption, which is equivalent in many respects to the activity of sellinginsurance, is potentially quite risky, given the large contingentliabilities potentially involved. Nonetheless, option selling iscommonplace in conventional, non-DBAR markets.

As indicated above, an advantage of the digital options representationof the DBAR methods of the present invention is the presentation of aninterface which displays bids and offers and therefore, by design,allows users to make investments in sets of DBAR contingent claims whoseP&L scenarios are comparable to those from traditional “sales” as wellas purchases of digital calls, puts, spreads, and strips. Specificallyin this context, “selling” entails the ability to achieve a profit andloss profile which is analogous to that achieved by sellers of digitaloptions instruments in non-DBAR markets, i.e., achieving a profit equalto the premium should the digital option expire out of the money, andsuffering a net loss equal to the digital option payout (or thenotional) less the premium received should the digital option expire inthe money.

In a preferred embodiment of a digital options exchange using the DBARcontingent claims methods and systems of the present invention, themechanics of “selling” involves converting such “sell” orders tocomplementary buy orders. Thus, a sale of the MSFT digital put optionswith strike price equal to 50, would be converted, in a preferred DBARDOE embodiment, to a complementary purchase of the 50 strike digitalcall options. A detailed explanation of the conversion process of a“sale” to a complementary buy order is provided in connection with thedescription of FIG. 15.

The complementary conversion of DBAR DOE “sales” to buys is facilitatedby interpreting the amount to be “sold” in a manner which is somewhatdifferent from the amount to be bought for a DBAR DOE buy order. In apreferred embodiment, when a trader specifies an amount in an order tobe “sold,” the amount is interpreted as the total amount of loss thatthe trader will suffer should the digital option, spread, or strip soldexpire in the money. As indicated above, the total amount lost or netloss is equal to the notional payout less the premium from the sale. Forexample, if the trader “sells” $1,000,000 of the MSFT digital put struckat 50, if the price of MSFT at expiration is 50 or below, then thetrader will lose $1,000,000. Correspondingly, in a preferred embodimentof the present invention, the order amount specified in a DBAR DOE“sell” order is interpreted as the net amount lost should the option,strip, or spread sold expire in the money. In conventional optionsmarkets, the amount would be interpreted and termed a “notional” or“notional amount” less the premium received, since the actual amountlost should the option expire in the money is the payout, or notional,less the premium received. By contrast, the amount of a buy order, in apreferred DBAR DOE embodiment, is interpreted as the amount to beinvested over the range of defined states which will generate the payoutshape or profile expected by the trader. The amount to be invested istherefore equivalent to the option “premium” in conventional optionsmarkets. Thus, in preferred embodiments of the present invention, forDBAR DOE buy orders, the order amount or premium is known and specifiedby the trader, and the contingent gain or payout should the optionpurchased finish in the money is not known until after all trading hasceased, the final equilibrium contingent claim “prices” or impliedprobabilities are calculated and any other termination criteria arefulfilled. By contrast, for a “sell” order in a preferred DBAR DOEembodiment of the present invention, the amount specified in the orderis the specified net loss (equal to the notional less the premium) whichrepresents the contingent loss should the option expire in the money.Thus, in a preferred embodiment, the amount of a buy order isinterpreted as an investment amount or premium which generates anuncertain payout until all predetermined termination criteria have beenmet; and the amount of a “sell” order is interpreted as a certain netloss should the option expire in the money corresponding to aninvestment amount or premium that remains uncertain until allpredetermined termination criteria have been met. In other words, in aDBAR DOE preferred embodiment, buy orders are for “premium” while “sell”orders are for net loss should the option expire in the money.

A relatively simple example illustrates the process, in a preferredembodiment of the present invention, of converting a “sale” of a DBARdigital option, strip, or spread to a complementary buy and the meaningof interpreting the amount of a buy order and “sell” order differently.Referring the MSFT example illustrated in Table 6.1.1 and Table 6.2.1above, assume that a trader has placed a market order (conditional orlimit orders are described in detail below) to “sell” the digital putwith strike price equal to 50. Ignoring transaction costs, the “price”of the 50 digital put option is equal to the sum of the implied stateprobabilities spanning the states where the option is in the money(i.e., (0,30], (30,40], and (40,50]) and is approximately 0.5596266.When the 50 put is in the money, the 50 call is out of the money andvice versa. Accordingly, the 50 digital call is “complementary” to the50 digital put. Thus, “selling” the 50 digital put for a given amount isequivalent in a preferred embodiment to investing that amount in thecomplementary call, and that amount is the net loss that would besuffered should the 50 digital put expire in the money (i.e., 50 andbelow). For example, if a trader places a market order to “sell”1,000,000 value units of the 50 strike digital put, this 1,000,000 valueunits are interpreted as the net loss if the digital put option expiresin the money, i.e., it corresponds to the notional payout loss plus thepremium received from the “sale.”

In preferred embodiments of the present investment, the 1,000,000 valueunits to be “sold” are treated as invested in the complementary50-strike digital call, and therefore are allocated according to themultistate allocation algorithm described in connection with thedescription of FIG. 13. The 1,000,000 value units are allocated inproportion to the value units previously allocated to the range ofstates comprising the 50-strike digital call, as indicated in Table6.2.2 above. Should the digital put expire in the money, the trader“selling” the digital put loses 1,000,000 value units, i.e., the traderloses the payout or notional less the premium. Should the digital putfinish out of the money, the trader will receive a payout approximatelyequal to 2,242,583.42 value units (computed by taking the total amountof value units invested, or 101,000,000, dividing by the new totalinvested in each state above 50 where the digital put is out of themoney, and multiplying by the corresponding state investment). Thepayout is the same regardless of which state above 50 occurs uponfulfillment of the termination criteria, i.e., the multistate allocationhas achieved the desired payout profile for a digital option. In thisillustration, the “sell” of the put will profit by 1,242,583.42 shouldthe option sold expire out of the money. This profit is equivalent tothe premium “sold.” On the other hand, to achieve a net loss of1,000,000 value units from a payout of 2,242,583.42, the premium is setat 1,242,583.42 value units.

The trader who “sells” in a preferred embodiment of a DBAR DOE specifiesan amount that is the payout or notional to be sold less the premium tobe received, and not the profit or premium to be made should the optionexpire out of the money. By specifying the payout or notional “sold”less the premium, this amount can be used directly as the amount to beinvested in the complementary option, strip, or spread. Thus, in apreferred embodiment, a DBAR digital options exchange can replicate orsynthesize the equivalent of trades involving the sale of option payoutsor notional (less the premium received) in the traditional market.

In another preferred embodiment, an investor may be able to specify theamount of premium to be “sold.” To illustrate this embodiment, quantityof premium to be “sold” can be assigned to the variable x. An investmentof quantity y on the states complementary to the range of states being“sold” is related to the premium x in the following manner:

${\frac{y}{1 - p} - y} = x$

where p is the final equilibrium “price”, including the “sale” x (andthe complementary investment y) of the option being “sold.” Rearrangingthis expression yields the amount of the complementary buy investment ythat must be made to effect the “sale” of the premium x:

$y = {x*\frac{\left( {1 - p} \right)}{p}}$

From this it can be seen that, given an amount of premium x that isdesired to be “sold,” the complementary investment that must be boughton the complement states in order for the trader to receive the premiumx, should the option “sold” expire out of the money, is a function ofthe price of the option being “sold.” Since the price of the optionbeing “sold” can be expected to vary during the trading period, in apreferred embodiment of a DBAR DOE of the present invention, the amounty required to be invested in the complementary state as a buy order canalso be expected to vary during the trading period.

In a preferred embodiment, traders may specify an amount of notionalless the premium to be “sold” as denoted by the variable y. Traders maythen specify a limit order “price” (see Section 6.8 below for discussionof limit orders) such that, by the previous equation relating y to x, atrader may indirectly specify a minimum value of x with the specifiedlimit order “price,” which may be substituted for p in the precedingequation. In another preferred embodiment, an order containingiteratively revised y amounts, as “prices” change during the tradingperiod are submitted. In another preferred embodiment, recalculation ofequilibrium “prices” with these revised y amounts is likely to lead to aconvergence of the y amounts in equilibrium. In this embodiment aniterative procedure may be employed to seek out the complementary buyamounts that must be invested on the option, strip, or spreadcomplementary to the range of states comprising the option being “sold”in order to replicate the desired premium that the trader desired to“sell.” This embodiment is useful since it aims to make the act of“selling” in a DBAR DOE more similar to the traditional derivativesmarkets.

It should be emphasized that the traditional markets differ from thesystems and methods of the present invention in as least one fundamentalrespect. In traditional markets, the sale of an option requires a sellerwho is willing to sell the option at an agreed-upon price. An exchangeof DBAR contingent claims of the present invention, in contrast, doesnot require or involve such sellers. Rather, appropriate investments maybe made (or bought) in contingent claims in appropriate states so thatthe payout to the investor is the same as if the claim, in a traditionalmarket, had been sold. In particular, using the methods and systems ofthe present invention, the amounts to be invested in various states canbe calculated so that the payout profile replicates the payout profileof a sale of a digital option in a traditional market, but without theneed for a seller. These steps are described in detail in connectionwith FIG. 15.

6.6 Clearing and Settlement

In a preferred embodiment of a digital options exchange using the DBARcontingent claims systems and methods of the present invention, alltypes of positions may be processed as digital options. This is becauseat fixing (i.e., the finalization of contingent claim “prices” orimplied probabilities at the termination of the trading period or otherfulfillment of all of the termination criteria) the profit and lossexpectations of all positions in the DBAR exchange are, from thetrader's perspective, comparable to if not the same as the profit andloss expectations of standard digital options commonly traded in the OTCmarkets, such as the foreign exchange options market (but without thepresence of actual sellers, who are needed on traditional optionsexchanges or in traditional OTC derivatives markets). The contingentclaims in a DBAR DOE of the present invention, once finalized at the endof a trading period, may therefore be processed as digital options orcombinations of digital options. For example, a MSFT digital option callspread with a lower strike of 40 and upper strike of 60 could beprocessed as a purchase of the lower strike digital option and a sale ofthe upper strike digital option.

There are many vendors of back office software that can readily handlethe processing of digital options. For example, Sungard, Inc., producesa variety of mature software systems for the processing of derivativessecurities, including digital options. Furthermore, in-house derivativessystems currently in use at major banks have basic digital optionscapability. Since digital options are commonly encountered instruments,many of the middleware initiatives currently underway e.g., FINXML, willlikely incorporate a standard protocol for handling digital options.Therefore, an advantage of a preferred embodiment of the DBAR DOE of thepresent invention is the ability to integrate with and otherwise useexisting technology for such an exchange.

6.7 Contract Initialization

Another advantage of the systems and methods of the present invention isthat, as previously noted, digital options positions can be representedinternally as composite trades. Composite trades are useful since theyhelp assure that an equilibrium distribution of investments among thestates can be achieved. In preferred embodiments, digital option andspreading activity will contribute to an equilibrium distribution. Thus,in preferred embodiments, indicative distributions may be used toinitialize trading at the beginning of the trading period.

In a preferred embodiment, these initial distributions may berepresented as investments or opening orders in each of the definedstates making up the contract or in the group of DBAR contingent claimsbeing traded in the auction. Since these investments need not be actualtrader investments, they may be reallocated among the defined states asactual trading occurs, so long as the initial investments do not changethe implicit probabilities of the states resulting from actualinvestments. In a preferred embodiment, the reallocation of initialinvestments is performed gradually so as to maximize the stability ofdigital call and put “prices” (and spreads), as viewed by investors. Bythe end of the trading period, all of the initial investments may bereallocated in proportion to the investments in each of the definedstates made by actual traders. The reallocation process may berepresented as a composite trade that has a same payout irrespective ofwhich of the defined states occurs. In preferred embodiments the initialdistribution can be chosen using current market indications from thetraditional markets to provide guidance for traders, e.g., optionsprices from traditional option markets can be used to calculate atraditional market consensus probability distribution, using forexample, the well-known technique of Breeden and Litzenberger. Otherreasonable initial and indicative distributions could be used.Alternatively, in a preferred embodiment, initialization can beperformed in such a manner that each defined state is initialized with avery small amount, distributed equally among each of the defined states.For example, each of the defined states could be initialized with 10⁻⁶value units. Initialization in this manner is designed to start eachstate with a quantity that is very small, distributed so as to provide avery small amount of information regarding the implied probability ofeach defined state. Other initialization methods of the defined statesare possible and could be implemented by one of skill in the art.

6.8 Conditional Investments, or Limit Orders

In a preferred embodiment of the system and methods of the presentinvention, traders may be able to make investments which are onlybinding if a certain “price” or implied probability for a given state ordigital option (or strip, spread, etc.) is achieved. In this context,the word “price,” is used for convenience and familiarity and, in thesystems and methods of the present invention, reflects the impliedprobability of the occurrence of the set of states corresponding to anoption—i.e., the implied probability that the option expires “in themoney.” For instance, in the example reflected in Table 6.2.1, a tradermay wish to make an investment in the MSFT digital call options withstrike price of 50, but may desire that such an investment actually bemade only if the final equilibrium “price” or implied probability is0.42 or less. Such a conditional investment, which is conditional uponthe final equilibrium “price” for the digital option, is sometimesreferred to (in conventional markets) as a “limit order.” Limit ordersare popular in traditional markets since they provide the means forinvestors to execute a trade at “their price” or better. Of course,there is no guarantee that such a limit order—which may be placedsignificantly away from the current market price—will in fact beexecuted. Thus, in traditional markets, limit orders provide the meansto control the price at which a trade is executed, without the traderhaving to monitor the market continuously. In the systems and method ofthe present invention, limit orders provide a way for investors tocontrol the likelihood that their orders will be executed at theirpreferred “prices” (or better), also without having continuously tomonitor the market.

In a preferred embodiment of a DBAR DOE, traders are permitted to buyand sell digital call and put options, digital spreads, and digitalstrips with limit “prices” attached. The limit “price” indicates that atrader desires that his trade be executed at that indicated limit“price”—actually the implied probability that the option will expire inthe money—“or better.” In the case of a purchase of a digital option,“better” means at the indicated limit “price” implied probability orlower (i.e., purchasing not higher than the indicated limit “price”). Inthe case of a “sale” of a DBAR digital option, “better” means at theindicated limit “price” (implied probability) or higher (i.e., sellingnot lower than the indicated limit “price”).

A benefit of a preferred embodiment of a DBAR DOE of the presentinvention which includes conditional investments or limit orders is thatthe placing of limit orders is a well-known mechanism in the financialmarkets. By allowing traders and investors to interact with a DBAR DOEof the present invention using limit orders, more liquidity should flowinto the DBAR DOE because of the familiarity of the mechanism, eventhough the underlying architecture of the DBAR DOE is different from theunderlying architecture of other financial markets.

The present invention also includes novel methods and systems forcomputing the equilibrium “prices” or implied probabilities, in thepresence of limit orders, of DBAR contingent claims in the variousstates. These methods and systems can be used to arrive at anequilibrium exclusively in the presence of limit orders, exclusively inthe presence of market orders, and in the presence of both. In apreferred embodiment, the steps to compute a DBAR DOE equilibrium for agroup of contingent claims including at least one limit order aresummarized as follows:

-   -   6.8(1) Convert all “sale” orders to complementary buy orders.        This is achieved by (i) identifying the states complementary to        the states being sold; (ii) using the amount “sold” as the        amount to be invested in the complementary states, and;        and (iii) for limit orders, adjusting the limit “price” to one        minus the original limit “price.”    -   6.8(2) Group the limit orders by placing all of the limit orders        which span or comprise the same range of defined states into the        same group. Sort each group from the best (highest “price” buy)        to the worst (lowest “price” buy). All orders may be processed        as buys since any “sales” have previously been converted to        complementary buys. For example, in the context of the MSFT        Digital Options illustrated in Table 6.2.1, there would be        separate groups for the 30 digital calls, the 30 digital puts,        the 40 digital calls, the 40 digital puts, etc. In addition,        separate groups are made for each spread or strip that spans or        comprises a distinct set of defined states.    -   6.8(3) Initialize the contract or group of DBAR contingent        claim. This may be done, in a preferred embodiment, by        allocating minimal quantities of value units uniformly across        the entire distribution of defined states so that each defined        state has a non-zero quantity of value units.    -   6.8(4) For all limit orders, adjust the limit “prices” of such        orders by subtracting from each limit order the order,        transaction or exchange fees for the respective contingent        claims.    -   6.8(5) With all orders broken into minimal size unit lots (e.g.,        one dollar or other small value unit for the group of DBAR        contingent claims), identify one order from a group that has a        limit “price” better than the current equilibrium “price” for        the option, spread, or strip specified in the order.    -   6.8(6) With the identified order, find the maximum number of        additional unit lots (“lots”) than can be invested such that the        limit “price” is no worse than the equilibrium “price” with the        chosen maximum number of unit lots added. The maximum number of        lots can be found by (i) using the method of binary search, as        described in detail below, (ii) trial addition of those lots to        already-invested amounts and (iii) recalculating the equilibrium        iteratively.    -   6.8(7) Identify any orders which have limit “prices” worse than        the current calculated equilibrium “prices” for the contract or        group of DBAR contingent claims. Pick such an order with the        worst limit “price” from the group containing the order. Remove        the minimum quantity of unit lots required so that the order's        limit “price” is no longer worse than the equilibrium “price”        calculated when the unit lots are removed. The number of lots to        be removed can be found by (i) using the method of binary        search, as described in detail below, (ii) trial subtraction of        those lots from already invested amounts and (iii) recalculating        the equilibrium iteratively.    -   6.8(8) Repeat steps 6.8(5) to 6.8(7). Terminate those steps when        no further additions or removals are necessary.    -   6.8(9) Optionally, publish the equilibrium from step 6.8(8) both        during the trading period and the final equilibrium at the end        of the trading period. The calculation during the trading period        is performed “as if” the trading period were to end at the        moment the calculation is performed. All prices resulting from        the equilibrium computation are considered mid-market prices,        i.e., they do not include the bid and offer spreads owing to        transaction fees. Published offer (bid) “prices” are set equal        to the mid-market equilibrium “prices” plus (minus) the fee.

In a preferred embodiment, the preceding steps 6.8(1) to 6.8(8) andoptionally step 6.8(9) are performed each time the set of orders duringthe trading or auction period changes. For example, when a new order issubmitted or an existing order is cancelled (or otherwise modified) theset of orders changes, steps 6.8(1) to 6.8(8) (and optionally step6.8(9)) would need to be repeated.

The preceding steps result in an equilibrium of the DBAR contingentclaims and executable orders which satisfy typical trader expectationsfor a market for digital options:

-   -   (1) At least some buy (“sell”) orders with a limit “price”        greater (less) than or equal to the equilibrium “price” for the        given option, spread or strip are executed or “filled.”    -   (2) No buy (“sell”) orders with limit “prices” less (greater)        than the equilibrium “price” for the given option, spread or        strip are executed.    -   (3) The total amount of executed lots equals the total amount        invested across the distribution of defined states.    -   (4) The ratio of payouts should each constituent state of a        given option, spread, or strike occur is as specified by the        trader, (including equal payouts in the case of digital        options), within a tolerable degree of deviation.    -   (5) Conversion of filled limit orders to customer orders for the        respective filled quantities and recalculating the equilibrium        does not materially change the equilibrium.    -   (6) Adding one or more lots to any of the filled limit orders        converted to market orders in step (5) and recalculating of the        equilibrium “prices” results in “prices” which violate the limit        “price” of the order to which the lot was added (i.e., no more        lots can be “squeaked in” without forcing market prices to go        above the limit “prices” of buy orders or below the limit        “prices” of sell orders).

The following example illustrates the operation of a preferredembodiment of a DBAR DOE of the present invention exclusively with limitorders. It is anticipated that a DBAR DOE will operate and process bothlimit and non-limit or market orders. As apparent to a person of skillin the art, if a DBAR DOE can operate with only limit orders, it canalso operate with both limit orders and market orders.

Like earlier examples, this example is also based on digital optionsderived from the price of MSFT stock. To reduce the complexity of theexample, it is assumed, for purposes of illustration, that there areillustrative purposes, only three strike prices: $30, $50, and $80.

TABLE 6.8.1 Buy Orders Limit Limit Limit “Price” Quantity “Price”Quantity “Price” Quantity 30 calls 50 calls 80 calls 0.82 10000 0.4310000 0.1 10000 0.835 10000 0.47 10000 0.14 10000 0.84 10000 0.5 1000080 puts 50 puts 30 puts 0.88 10000 0.5 10000 0.16 10000 0.9 10000 0.5210000 0.17 10000 0.92 10000 0.54 10000

TABLE 6.8.2 “Sell” Orders Limit Limit Limit “Price” Quantity “Price”Quantity “Price” Quantity 30 calls 50 calls 80 calls 0.81 5000 0.4210000 0.11 10000 0.44 10000 0.12 10000 80 puts 50 puts 30 puts 0.9 200000.45 10000 0.15 5000  0.50 10000 0.16 10000The quantities entered in the “Sell Orders” table, Table 6.8.2, are thenet loss amounts which the trader is risking should the option “sold”expire in the money, i.e., they are equal to the notional less thepremium received from the sale, as discussed above.

-   -   (i) According to step 6.8(1) of the limit order methodology        described above, the “sale” orders are first converted to buy        orders. This involves switching the contingent claim “sold” to a        buy of the complementary contingent claim and creating a new        limit “price” for the converted order equal to one minus the        limit “price” of the sale. Converting the “sell” orders in Table        6.8.2 therefore yields the following converted buy orders:

TABLE 6.8.3 “Sale” Orders Converted to Buy Orders Limit Limit Limit“Price” Quantity “Price” Quantity “Price” Quantity 30 puts 50 puts 80puts 0.19  5000 0.58 10000 0.89 10000 0.56 10000 0.88 10000 80 calls 50calls 30 calls 0.1 20000 0.55 10000 0.85  5000 0.50 10000 0.84 10000

-   -   (ii) According to step 6.8(2), the orders are then placed into        groupings based upon the range of states which each underlying        digital option comprises or spans. The groupings for this        illustration therefore are: 30 calls, 50 calls, 80 calls, 30        puts, 50 puts, and 80 puts    -   (iii) In this illustrative example, the initial liquidity in        each of the defined states is set at one value unit.    -   (iv) According to step 6.8(4), the orders are arranged from        worst “price” (lowest for buys) to best “price” (highest for        buys). Then, the limit “prices” are adjusted for the effect of        transaction or exchange costs. Assuming that the transaction fee        for each order is 5 basis points (0.0005 value units), then        0.0005 is subtracted from each limit order price. The aggregated        groups for this illustrative example, sorted by adjusted limit        prices (but without including the initial one-value-unit        investments), are as displayed in the following table:

TABLE 6.8.4 Aggregated, Sorted, Converted, and Adjusted Limit OrdersLimit Limit Limit “Price” Quantity “Price” Quantity “Price” Quantity 30calls 50 calls 80 calls 0.8495  5000 0.5495 10000 0.1395 10000 0.839520000 0.4995 20000 0.0995 30000 0.8345 10000 0.4695 10000 0.8195 100000.4295 10000 80 puts 50 puts 30 puts 0.9195 10000 0.5795 10000 0.1895 5000 0.8995 10000 0.5595 10000 0.1695 10000 0.8895 10000 0.5395 100000.1595 10000 0.8795 20000 0.5195 10000 0.4995 10000

-   -   -   After adding the initial liquidity of one value unit in each            state, the initial option prices are as follows:

TABLE 6.8.5 MSFT Digital Options Initial Prices CALLS PUTS IND IND INDIND IND IND STRIKE MID BID OFFER MID BID OFFER 30 0.85714 0.856640.85764 0.14286 0.14236 0.14336 50 0.57143 0.57093 0.57193 0.428570.42807 0.42907 80 0.14286 0.14236 0.14336 0.85714 0.85664 0.85764

-   -   (v) According to step 6.8(5) and based upon the description of        limit order processing in connection with FIG. 12, in this        illustrative example an order from Table 6.8.4 is identified        which has a limit “price” better or higher than the current        market “price” for a given contingent claim. For example, from        Table 6.9.4, there is an order for 10000 digital puts struck at        80 with limit “price” equal to 0.9195. The current mid-market        “price” for such puts is equal to 0.85714.    -   (vi) According to step 6.8(6), by the methods described in        connection with FIG. 17, the maximum number of lots of the order        for the 80 digital puts is added to already-invested amounts        without increasing the recalculated mid-market “price,” with the        added lots, above the limit order price of 0.9195. This process        discovers that, when five lots of the 80 digital put order for        10000 lots and limit “price” of 0.9195 are added, the new        mid-market price is equal to 0.916667. Assuming the distribution        of investments for this illustrative example, addition of any        more lots will drive the mid-market price above the limit price.        With the addition of these lots, the new market prices are:

TABLE 6.8.5 MSFT Digital Options Prices after addition of five lots of80 puts CALLS PUTS IND IND IND IND IND IND STRIKE MID BID OFFER MID BIDOFFER 30 0.84722 0.84672 0.84772 0.15278 0.15228 0.15328 50 0.541670.54117 0.54217 0.45833 0.45783 0.45883 80 0.08333 0.08283 0.083830.91667 0.91617 0.91717

-   -   -   As can be seen from Table 6.8.5, the “prices” of the call            options have decreased while the “prices” of the put options            have increased as a result of filling five lots of the 80            digital put options, as expected.

    -   (vii) According to step 6.8(7), the next step is to determine,        as described in FIG. 17, whether there are any limit orders        which have previously been filled whose limit “prices” are now        less than the current mid-market “prices,” and as such, should        be subtracted. Since there are no orders than have been filled        other than the just filled 80 digital put, there is no removal        or “prune” step required at this stage in the process.

    -   (viii) According to step 6.8(8), the next step is to identify        another order which has a limit “price” higher than the current        mid-market “prices” as a candidate for lot addition. Such a        candidate is the order for 10000 lots of the 50 digital puts        with limit price equal to 0.5795. Again the method of binary        search is used to determine the maximum number of lots that can        be added from this order to already-invested amounts without        letting the recalculated mid-market “price” exceed the order's        limit price of 0.5795. Using this method, it can be determined        that only one lot can be added without forcing the new market        “price” including the additional lot above 0.5795. The new        prices with this additional lot are then:

TABLE 6.8.6 MSFT Digital Options “Prices” after (i) addition of fivelots of 80 puts and (ii) addition of one lot of 50 puts CALLS PUTS INDIND IND IND IND IND STRIKE MID BID OFFER MID BID OFFER 30 0.824200.82370 0.82470 0.17580 0.17530 0.17630 50 0.47259 0.47209 0.473090.52741 0.52691 0.52791 80 0.07692 0.07642 0.07742 0.923077 0.922580.92358

-   -   -   Continuing with step 6.8(8), the next step is to identify an            order whose limit “price” is now worse (i.e., lower than)            the mid-market “prices” from the most recent equilibrium            calculation as shown in Table 6.8.6. As can be seen from the            table, the mid-market “price” of the 80 digital put options            is now 0.923077. The best limit order (highest “priced”) is            the order for 10000 lots at 0.9195, of which five are            currently filled. Thus, the binary search routine determines            the minimum number of lots which are to be removed from this            order so that the order's limit “price” is no longer worse            (i.e., lower than) the newly recalculated market “price.”            This is the removal or prune part of the equilibrium            calculation.        -   The “add and prune” steps are repeated iteratively with            intermediate multistate equilibrium allocations performed.            The contract is at equilibrium when no further lots may be            added for orders with limit order “prices” better than the            market or removed for limit orders with “prices” worse than            the market. At this point, the group of DBAR contingent            claims (sometimes referred to as the “contract”) is in            equilibrium, which means that all of the remaining            conditional investments or limit orders—i.e., those that did            not get removed—receive “prices” in equilibrium which are            equal to or better than the limit “price” conditions            specified in each order. In the present illustration, the            final equilibrium “prices” are:

TABLE 6.8.7 MSFT Digital Options Equilibrium Prices CALLS PUTS STRIKEIND MID IND BID IND OFFER IND MID IND BID IND OFFER 30 0.830503 0.8300030.831003 0.169497 0.168997 0.169997 50 0.480504 0.480004 0.4810040.519496 0.518996 0.519996 80 0.139493 0.138993 0.139993 0.8605070.860007 0.861007

-   -   -   Thus, at these equilibrium “prices,” the following table            shows which of the original orders are executed or “filled”:

TABLE 6.8.8 Filled Buy Orders Limit “Price” Quantity Filled Limit“Price” Quantity Filled Limit “Price” Quantity Filled 30 calls 50 calls80 calls 0.82 10000 0 0.43 10000 0 0.1 10000 0 0.835 10000 10000 0.4710000 0 0.14 10000 8104 0.84 10000 10000 0.5 10000 10000 80 puts 50 puts30 puts 0.88 10000 10000 0.5 10000 0 0.16 10000 0 0.9 10000 10000 0.5210000 2425 0.17 10000 2148 0.92 10000 10000 0.54 10000 10000

TABLE 6.8.9 Filled Sell Orders Limit “Price” Quantity Filled Limit“Price” Quantity Filled Limit “Price” Quantity Filled 30 calls 50 calls80 calls 0.81  5000 5000 0.42 10000 10000 0.11 10000 10000 0.44 1000010000 0.12 10000 10000 80 puts 50 puts 30 puts 0.9 20000   0 0.45 1000010000 0.15  5000  5000 0.50 10000 10000 0.16 10000 10000

It may be possible only partially to execute or “fill” a trader's orderat a given limit “price” or implied probability of the relevant states.For example, in the current illustration, the limit buy order for 50puts at limit “price” equal to 0.52 for an order amount of 10000 may beonly filled in the amount 2424 (see Table 6.8.8). If orders are made bymore than one investor and not all of them can be filled or executed ata given equilibrium, in preferred embodiments it is necessary to decidehow many of which investor's orders can be filled, and how many of whichinvestor's orders will remain unfulfilled at that equilibrium. This maybe accomplished in several ways, including by filling orders on afirst-come-first-filled basis, or on a pro rata or other basis known orapparent to one of skill in the art. In preferred embodiments, investorsare notified prior to the commencement of a trading period about thebasis on which orders are filled when all investors' limit orders cannotbe filled at a particular equilibrium.

6.9 Sensitivity Analysis and Depth of Limit Order Book

In preferred embodiments of the present invention, traders in DBARdigital options may be provided with information regarding the quantityof a trade that could be executed (“filled”) at a given limit “price” orimplied probability for a given option, spread or strip. For example,consider the MSFT digital call option with strike of 50 illustrated inTable 6.1.1 above. Assume the current “price” or implied probability ofthe call option is 0.4408 on the “offer” side of the market. A tradermay desire, for example, to know what quantity of value units could betransacted and executed at any given moment for a limit “price” which isbetter than the market. In a more specific example, for a purchase ofthe 50 strike call option, a trader may want to know how much would befilled at that moment were the trader to specify a limit “price” orimplied probably of, for example, 0.46. This information is notnecessarily readily apparent, since the acceptance of conditionalinvestments (i.e., the execution of limit orders) changes the impliedprobability or “price” of each of the states in the group. As the limit“price” is increased, the quantities specified in a buy order are morelikely to be filled, and a curve can be drawn with the associated limit“price”/quantity pairs. The curve represents the amount that could befilled (for example, along the X-axis) versus the corresponding limit“price” or implied probability of the strike of the order (for example,along the Y-axis). Such a curve should be useful to traders, since itprovides an indication of the “depth” of the DBAR DOE for a givencontract or group of contingent claims. In other words, the curveprovides information on the “price” or implied probability, for example,that a buyer would be required to accept in order to execute apredetermined or specified number of value units of investment for thedigital option.

6.10 Networking of DBAR Digital Options Exchanges

In preferred embodiments, one or more operators of two or more differentDBAR Digital Options Exchanges may synchronize the time at which tradingperiods are conducted (e.g., agreeing on the same commencement andpredetermined termination criteria) and the strike prices offered for agiven underlying event to be observed at an agreed upon time. Eachoperator could therefore be positioned to offer the same trading periodon the same underlying DBAR event of economic significance or financialinstrument. Such synchronization would allow for the aggregation ofliquidity of two or more different exchanges by means of computing DBARDOE equilibria for the combined set of orders on the participatingexchanges. This aggregation of liquidity is designed to result in moreefficient “pricing” so that implied probabilities of the various statesreflect greater information about investor expectations than if a singleexchange were used.

7. DBAR DOE: ANOTHER EMBODIMENT

In another embodiment of a DBAR Digital Options Exchange (“DBAR DOE”), atype of demand-based market or auction, all orders for digital optionsare expressed in terms of the payout (or “notional payout”) receivedshould any state of the set of constituent states of a DBAR digitaloption occur (as opposed to, for example, expressing buy digital optionorders in terms of premium to be invested and expressing “sell” digitaloption orders in terms of notional payout, or notional payout less thepremium received). In this embodiment, the DBAR DOE can accept andprocess limit orders for digital options expressed in terms of eachtrader's desired payout. In this embodiment, both buy and sell ordersmay be handled consistently, and the speed of calculation of theequilibrium calculation is increased. This embodiment of the DBAR DOEcan be used with or without limit orders (also referred to asconditional investments). Additionally this embodiment of the DBAR DOEcan be used to trade in a demand-based market or auction based on anyevent, regardless of whether the event is economically significant ornot.

In this embodiment, an equilibrium algorithm (set forth in Equations7.3.7 and 7.4.7) may be used on orders without limits (without limits onthe price), to determine the prices and total premium invested into aDBAR DOE market or auction based only upon information concerning therequested payouts per order and the defined states (or spreads) forwhich the desired digital option is in-the-money (the payout profile forthe order). The requested payout per order is the executed notionalpayout per order, and the trader or user pays the price determined atthe end of the trading period by the equilibrium algorithm necessary toreceive the requested payout.

In this embodiment, an optimization system (also referred to as theOrder Price Function or OPF) may also be utilized that maximizes thepayouts per order within the constraints of the limit order. In otherwords, when a user or trader specifies a limit order price, and alsospecifies the requested payouts per order and the defined states (orspreads) for which the desired digital option is in-the-money, then theoptimization system or OPF determines a price of each order that is lessthan or equal to each order's limit price, while maximizing the executednotional payout for the orders. As set forth below, in this limit orderexample, the user may not receive the requested payout but will receivea maximum executed notional payout given the limit price that the userdesires to invest for the payout.

In other words, in this embodiment, three mathematical principlesunderlie demand-based markets or auctions: demand-based pricing andself-funding conditions; how orders in digital options are constitutedin a demand-based market or auction; and, how a demand-based auction ormarket may be implemented with standard limit orders. Similarequilibrium algorithms, optimization systems, and mathematicalprinciples also underlie and apply to demand-based markets or auctionsthat include one or more customer orders for derivatives strategies orother contingent claims, that are replicated or approximated with a setof replicating claims, which can be digital options and/or vanillaoptions, as described in greater detail in Sections 10, 11 and 13 below.These customer orders are priced based upon a demand-based valuation ofthe replicating digital options and/or vanilla options that replicatethe derivatives strategies, and the demand-based valuation includes theapplication of the equilibrium algorithm, optimization system andmathematical principals to such an embodiment.

In this DBAR DOE embodiment, for each demand-based market or auction,the demand-based pricing condition applies to every pair of fundamentalcontingent claims. In demand-based systems, the ratio of prices of eachpair of fundamental contingent claims is equal to the ratio of volumefilled for those claims. This is a notable feature of DBAR contingentclaims markets because the demand-based pricing condition relates theamount of relative volumes that may clear in equilibrium to the relativeequilibrium market prices. Thus, a demand-based market microstructure,which is the foundation of demand-based market or auction, is uniqueamong market mechanisms in that the relative prices of claims aredirectly related to the relative volume transacted of those claims. Bycontrast, in conventional markets, which have heretofore not adopteddemand-based principles, relative contingent claim prices typicallyreflect, in theory, the absence of arbitrage opportunities between suchclaims, but nothing is implied or can be inferred about the relativevolumes demanded of such claims in equilibrium.

Equation 7.4.7, as set forth below, is the equilibrium equation fordemand-based trading in accordance with one embodiment of the presentinvention. It states that a demand-based trading equilibrium can bemathematically expressed in terms of a matrix eigensystem, in which thetotal premium collected in a demand-based market or auction (T) is equalto the maximum eigenvalue of a matrix (H) which is a function of theaggregate notional amounts executed for each fundamental spread and theopening orders. In addition, the eigenvector corresponding to thismaximum eigenvalue, when normalized, contains the prices of thefundamental single strike spreads. Equation 7.4.7 shows that givenaggregate notional amounts to be executed (Y) and arbitrary amounts ofopening orders (K), that a unique demand-based trading equilibriumresults. The equilibrium is unique because a unique total premiuminvestment, T, is associated with a unique vector of equilibrium prices,p, by the solution of the eigensystem of Equation 7.4.7.

Demand-based markets or auctions may be implemented with a standardlimit order book in which traders attach price conditions for executionof buy and sell orders. As in any other market, limit orders allowtraders to control the price at which their orders are executed, at therisk that the orders may not be executed in full or in part. Limitorders may be an important execution control feature in demand-basedauctions or markets because final execution is delayed until the end ofthe trading or auction period.

Demand-based markets or auctions may incorporate standard limit ordersand limit order book principles. In fact, the limit order book employedin a demand-based market or auction and the mathematical expressionsused therein may be compatible with standard limit order book mechanismsfor other existing markets and auctions. The mathematical expression ofa General Limit Order Book is an optimization problem in which themarket clearing solution to the problem maximizes the volume of executedorders subject to two constraints for each order in the book. Accordingto the first constraint, should an order be executed, the order's limitprice is greater than or equal to the market price including theexecuted order. According to the second constraint, the order's executednotional amount is not to exceed the notional amount requested by thetrader to be executed.

7.1 Special Notation

For the purposes of the discussion of the embodiment described in thepresent section, the following notation is utilized. The notation usessome symbols previously employed in other sections of thisspecification. It should be understood that the meanings of thesenotational symbols are valid as defined below only in the context of thediscussion in the present section (Section 7—DBAR DOE: ANOTHEREMBODIMENT as well as the discussion in relation to FIG. 19 and FIG. 20in Section 9).

Known Variables

-   m: number of defined states or spreads, a natural number. Index    letter i, i=1, 2, . . . m.-   k: m×1 vector where k_(i) is the initial invested premium for state    i, i=1, 2, . . . , m, k_(i) is a natural number so k_(i)>0 i=1, 2, .    . . , m-   e: a vector of ones of length m (m×1 unit vector)-   n: number of orders in the market or auction, a natural number.    Index letter j, j=1, 2, . . . , n-   r: n×1 vector where r_(j) is equal to the requested payout for order    j, j=1, 2, . . . , n. r_(j) is a natural number so r_(j) is positive    for all j, j=1, 2, . . . , n-   w: n×1 vector where w_(j) equals the inputted limit price for order    j, j=1, 2, . . . , n    -   Range: 0<w_(j)≦1 for j=1, 2, . . . , n for digital options        -   0<w_(j) for j=1, 2, . . . , n for arbitrary payout options-   w_(j) ^(a): n×1 vector where w_(j) ^(a) is the adjusted limit price    for order j after converting “sell” orders into buy orders (as    discussed below) and after adjusting the inputted limit order w_(j)    with fee f_(j) (assuming flat fee) for order j, j=1, 2 . . . , n    -   For a “sell” order j, the adjusted limit price w_(j) ^(a) equals        (1−w_(j)−f_(j))    -   For a buy order j, the adjusted limit price w_(j) ^(a) equals        (w_(j)−f_(j))-   B: n×m matrix where B_(j,i) is a positive number if the jth order    requests a payout for the i^(th) state, and 0 otherwise. For digital    options, the positive number is one.    -   Each row j of B comprises a payout profile for order j.-   f_(j): transaction fee for order j, scalar (in basis points) added    to and subtracted from equilibrium price to obtain offer and bid    prices, respectively, and subtracted from and added to limit prices,    w_(j), to obtain adjusted limit price, w_(j) ^(a) for buy and sell    limit prices, respectively.

Unknown Variables

-   x: n×1 vector where x_(j) is the notional payout executed for order    j in equilibrium    -   Range: 0≦x_(j)≦r_(j) for j=1, 2, . . . , n-   y: m×1 vector where y_(i) is the notional payout executed per    defined state i, i=1, 2, . . . , m    -   Definition: y≡B^(T)x-   T: positive scalar, not necessarily an integer.    -   T is the total invested premium (in value units) in the contract

$T = {{{\sum\limits_{i = 1}^{m}{y_{i}p_{i}}} + {\sum\limits_{i = 1}^{m}k_{i}}} = {{\sum\limits_{j = 1}^{n}{x_{j}\pi_{j}}} + {\sum\limits_{i = 1}^{m}k_{i}}}}$

-   T_(i): positive scalar, not necessarily an integer    -   T_(i) is the total invested premium (in value units) in state i-   p: m×1 vector where p_(i) is the price/probability for state i, i=1,    2, . . . , m

$p_{i} \equiv \frac{k_{i}}{T - y_{i}}$

-   π_(j): equilibrium price for order j-   π(x): B*p, an n×1 vector containing the equilibrium prices for each    order j.-   g: n×1 vector whose j element is g_(j) for j=1, 2, . . . , n    -   Definition: g≡B*p−w    -   Note B*p is the vector of market prices for order j denoted by        π_(j) g is the difference between the market prices and the        limit prices

7.2 Elements of Example DBAR DOE Embodiment

In this embodiment (Section 7), traders submit orders during the DBARmarket or auction that include the following data: (1) an order payoutsize (denoted r_(j)), (2) a limit order price (denoted w_(j)), and (3)the defined states for which the desired digital option is in-the-money(denoted as the rows of the matrix B, as described in the previoussub-section). In this embodiment, all of the order requests are in theform of payouts to be received should the defined states over which therespective options are in-the-money occur. In Section 6, an embodimentwas described in which the order amounts are invested premium amounts,rather than the aforementioned payouts.

7.3 Mathematical Principles

In this embodiment of a DBAR DOE market or auction, traders are able tobuy and sell digital options and spreads. The fundamental contingentclaims of this market or auction are the smallest digital optionspreads, i.e., those that span a single strike price. For example, ademand-based market or auction, such as, for example, a DBAR auction ormarket, that offers digital call and put options with strike prices of30, 40, 50, 60, and 70 contains six fundamental states: the spread belowand including 30; the spread between 30 and 40 including 40; the spreadbetween 40 and 50 including 50; etc. As indicated in the previoussection, in this embodiment, p_(i) is the price of a single strikespread i and m is the number of fundamental single state spreads or“defined states.” For these single strike spreads, the followingassumptions are made:

DBAR DOE Assumptions for this Embodiment

$\begin{matrix}{7.3{.1}} & \; \\{{\sum\limits_{i = 1}^{m}p_{i}} = 1} & (1) \\{{{p_{i} > {0\mspace{14mu} {for}\mspace{14mu} i}} = 1},\ldots \mspace{14mu},m} & (2) \\{{{k_{i} > {0\mspace{14mu} {for}\mspace{14mu} i}} = 1},\ldots \mspace{14mu},m} & (3)\end{matrix}$

The first assumption, equation 7.3.1(1), is that the fundamental spreadprices sum to unity. This equation holds for this embodiment as well asfor other embodiments of the present invention. Technically, the sum ofthe fundamental spread prices should sum to the discount factor thatreflects the time value of money (i.e., the interest rate) prevailingfrom the time at which investors must pay for their digital options tothe time at which investors receive a payout from an in-the-money optionafter the occurrence of a defined state. For the purposes of thisdescription of this embodiment, the time value of money during thisperiod will be taken to be zero, i.e., it will be ignored so that thefundamental spread prices sum to unity. The second assumption, equation7.3.1(2), is that each price must be positive. Assumption 3, equation7.3.1(3), is that the DBAR DOE contract of the present embodiment isinitialized (see Section 6.7, above) with value units invested in eachstate in the amount of k_(i) (initial amount of value units invested forstate i).

Using the notation from Section 7.1, the Demand Reallocation Function(DRF) of this embodiment of an OPF is a canonical DRF (CDRF), settingthe total amount of investments that are allocated using multistateallocation techniques to the defined states equal to the total amount ofinvestment in the auction or market that is available (net of anytransaction fees) to allocate to the payouts upon determining thedefined state which has occurred. Alternatively, a non-canonical DRF maybe used in an OPF.

Under a CDRF, the total amount invested in each defined state is afunction of the price in that state, the total amount of notional payoutrequested for that state, and the initial amount of value units investedin the defined state, or:

T _(i) =p _(i) *y _(i) +k _(i)  7.3.2

The ratio of the invested amounts in any two states is therefore equalto:

$\begin{matrix}{\frac{T_{i}}{T_{j}} = \frac{{p_{i}*y_{i}} + k_{i}}{{p_{j}*y_{j}} + k_{j}}} & {7.3{.3}}\end{matrix}$

As described previously, since each state price is equal to the totalinvestment in the state divided by the total investment over all of thestates (p_(i)=T_(i)/T and p_(j)=T_(j)/T), the ratio of the investmentamounts in each DBAR contingent claim defined state is equal to theratio of the prices or implied probabilities for the states, which,using the notation of Section 7.1, yields:

$\begin{matrix}{\frac{T_{i}}{T_{j}} = {\frac{{p_{i}*y_{i}} + k_{i}}{{p_{j}*y_{j}} + k_{j}} = \frac{p_{i}}{p_{j}}}} & {7.3{.4}}\end{matrix}$

Eliminating the denominators of the previous equation and summing over jyields:

$\begin{matrix}{{\sum\limits_{j = 1}^{m}{p_{j}\left( {{p_{i}*y_{i}} + k_{i}} \right)}} = {\sum\limits_{j = 1}^{m}{p_{i}*\left( {{p_{j}*y_{j}} + k_{j}} \right)}}} & {7.3{.5}}\end{matrix}$

Substitution for T into the above equation yields:

$\begin{matrix}{{\left( {{p_{i}*y_{i}} + k_{i}} \right)\left( {\sum\limits_{j = 1}^{m}p_{j}} \right)} = {p_{i}T}} & {7.3{.6}}\end{matrix}$

By the assumption that the state prices or probabilities sum to unityfrom Equation 7.3.1, this yields the following equation:

$\begin{matrix}{p_{i} = \frac{k_{i}}{T - y_{i}}} & {7.3{.7}}\end{matrix}$

This equation yields the state price or probability of a defined statein terms of: (1) the amount of value units invested in each state toinitialize the DBAR auction or market (k_(i)); (2) the total amount ofpremium invested in the DBAR auction or market (T); and (3) the totalamount of payouts to be executed for all of the traders' orders forstate i (y_(i)). Thus, in this embodiment, Equation 7.3.7 follows fromthe assumptions stated above, as indicated in the equations in 7.3.1,and the requirement the DRF imposes that the ratio of the state pricesfor any two defined states in a DBAR auction or market be equal to theratio of the amount of invested value units in the defined states, asindicated in Equation 7.3.4.

7.4 Equilibrium Algorithm

From equation 7.3.7 and the assumption that the probabilities of thedefined states sum to one (again ignoring any interest rateconsiderations), the following m+1 equations may be solved to obtain theunique set of defined state probabilities (p's) and the total premiuminvestment for the group of defined states or contingent claims:

$\begin{matrix}{7.4{.1}} & \; \\{{p_{i} = \frac{k_{i}}{T - y_{i}}},\mspace{14mu} {i = 1},2,\ldots \mspace{14mu},m} & (a) \\{{\sum\limits_{i = 1}^{m}p_{i}} = {{\sum\limits_{i = 1}^{m}\frac{k_{i}}{T - y_{i}}} = 1}} & (b)\end{matrix}$

Equation 7.4.1 contains m+1 unknowns and m+1 equations. The unknowns arethe p_(i), i=1, 2, . . . , m, and T, the total investment for all of thedefined states. In accordance with the embodiment, the method ofsolution of the m+1 equations is to first solve Equation 7.4.1 (b). Thisequation is a polynomial in T. By the assumption that all of theprobabilities of the defined states must be positive, as stated inEquation 7.3.1, and that the probabilities also sum to one, as alsostated in Equation 7.3.1, the defined state probabilities are between 0and 1 or:

$\begin{matrix}{{{0 < p_{i} < 1},{{which}\mspace{14mu} {implies}}}{{0 < \frac{k_{i}}{T - y_{i}} < 1},{{{for}\mspace{14mu} i} = 1},2,{\ldots \mspace{14mu} m},{{which}\mspace{14mu} {implies}}}{{T > {y_{i} + k_{i}}},{{{for}\mspace{14mu} i} = 1},2,{\ldots \mspace{14mu} m},{{which}\mspace{14mu} {implies}}}{{T > {\max \left( {y_{i} + k_{i}} \right)}},{{{for}\mspace{14mu} i} = 1},2,{\ldots \mspace{14mu} m}}} & {7.4{.2}}\end{matrix}$

So the lower bound for T is equal to:

T _(lower)=max(y _(i) +k _(i))

By Equation 7.3.2:

$\begin{matrix}{T = {{\sum\limits_{i = 1}^{m}T_{i}} = {{\sum\limits_{i = 1}^{m}k_{i}} + {\sum\limits_{i = 1}^{m}{p_{i}y_{i}}}}}} & {7.4{.3}}\end{matrix}$

Letting y_((m)) be the maximum value of the y's,

$\begin{matrix}\begin{matrix}{T = {{{\sum\limits_{i = 1}^{m}k_{i}} + {\sum\limits_{i = 1}^{m}{p_{i}y_{i}}}} \leq {{\sum\limits_{i = 1}^{m}k_{i}} + {\sum\limits_{i = 1}^{m}{p_{i}y_{(m)}}}}}} \\{= {{\sum\limits_{i = 1}^{m}k_{i}} + {y_{(m)}{\sum\limits_{i = 1}^{m}p_{i}}}}} \\{= {{\sum\limits_{i = 1}^{m}k_{i}} + y_{(m)}}}\end{matrix} & {7.4{.4}}\end{matrix}$

Thus, the upper bound for T is equal to:

$\begin{matrix}{T_{upper} = {{{\sum\limits_{i = 1}^{m}k_{i}} + y_{(m)}} = {{\max \left( y_{i} \right)} + {\sum\limits_{i = 1}^{m}k_{i}}}}} & {7.4{.5}}\end{matrix}$

The solution for the total investment in the defined states thereforelies in the following interval

$\begin{matrix}{{{T_{lower} < T \leq T_{upper}},{or}}{{\max \left( {y_{i} + k_{i}} \right)} < T \leq {{\max \left( y_{i} \right)} + {\sum\limits_{i = 1}^{m}k_{i}}}}} & {7.4{.6}}\end{matrix}$

In this embodiment, T is determined uniquely from the equilibriumexecution order amounts, denoted by the vector x. Recall that in thisembodiment, y≡B^(T)x. As shown above,

Tε(T_(lower),T_(upper)]

Let the function f be

${f(T)} = {{{\sum\limits_{i = 1}^{m}\left( \frac{k_{i}}{T - y_{i}} \right)} - 1} = {0 = {{\sum\limits_{i = 1}^{m}p_{i}} - 1}}}$

Further,

f(T _(lower))>0

f(T _(upper))<0

Now, over the range Tε(T_(rower), T_(upper)], f(T) is differentiable andstrictly monotonically decreasing. Thus, there is a unique T in therange such that

f(T)=0

Thus, T is uniquely determined by the x_(j)'s (the equilibrium executednotional payout amounts for each order j).

The solution for Equation 7.4.1(b) can therefore be obtained usingstandard root-finding techniques, such as the Newton-Raphson technique,over the interval for T stated in Equation 7.4.6. Recall that thefunction f(T) is defined as

${f(T)} = {{\sum\limits_{i = 1}^{m}\left( \frac{k_{i}}{T - y_{i}} \right)} - 1}$

The first derivative of this function is therefore:

${f^{\prime}(T)} = {\frac{f}{T} = {- {\sum\limits_{i = 1}^{m}\frac{k_{i}}{\left( {T - y_{i}} \right)^{2}}}}}$

Thus for T, take for an initial guess

T ⁰=Max(y ₁ +k ₁ ,y ₂ +k ₂, . . . , y _(m) +k _(m))

For the p+1^(st) guess use

$T^{p + 1} = {T^{p} - \frac{f\left( T^{p} \right)}{f^{\prime}\left( T^{p} \right)}}$

and calculate iteratively until a desired level of convergence to theroot of f(T), is obtained.

Once the solution for Equation 7.4.1(b) is obtained, the value of T canbe substituted into each of the m equations in 7.4.1(a) to solve for thep_(i). When the T and the p_(i) are known, all prices for DBAR digitaloptions and spreads may be readily calculated, as indicated by thenotation in 7.1.

Note that, in the alternative embodiment with no limit orders (brieflydiscussed at the beginning of this section 7), there are no constraintsset by limit prices, and the above equilibrium algorithm is easilycalculated because x_(j), the executed notional payout amounts for eachorder j, is equal to r_(j) (a known quantity), the requested notionalpayout for order j.

Regardless of the presence of limit orders, an equivalent set ofmathematics for this embodiment of a DBAR DOE is developed using matrixnotation. The matrix equivalent of Equation 7.3.2 may be written asfollows:

H*p=T*p  7.4.7

where T and p are the total premium and state probability vector,respectively, as described in Section 7.1. The matrix H, which has mrows and m columns where m is the number of defined states in the DBARmarket or auction, is defined as follows:

$\begin{matrix}{H = \begin{bmatrix}{y_{1} + k_{1}} & k_{1} & k_{1} & \ldots & k_{1} \\k_{2} & {y_{2} + k_{2}} & k_{2} & \ldots & k_{2} \\\vdots & \vdots & \vdots & \ldots & \vdots \\k_{m} & k_{m} & k_{m} & \ldots & {y_{m} + k_{m}}\end{bmatrix}} & {7.4{.8}}\end{matrix}$

H is a matrix with m rows and m columns. Each diagonal entry of H isequal to y_(i)+k_(i) (the sum of the notional payout requested by allthe traders for state i and the initial amount of value units investedfor state i). The other entries for each row are equal to k_(i) (theinitial amount of value units invested for state i). Equation 7.4.7 isan eigenvalue problem, where:

H=Y+K*V

-   -   Y=an m×m diagonal matrix of the aggregate notional amounts to be        executed, Y_(i,i)=y_(i)    -   K=an m×m diagonal matrix of the arbitrary amounts of opening        orders, K_(i,i)=k_(i)    -   V=an m×m matrix of ones, V_(i,j)=1    -   T=max (λ_(i)(H)), i.e., the maximum eigenvalue of the matrix H;        and    -   p=|ν(H,T)|, i.e., the normalized eigenvector associated with the        eigenvalue T.

Thus, Equation 7.4.7 is, in this embodiment, a method of mathematicallydescribing the equilibrium of a DBAR digital options market or auctionthat is unique given the aggregate notional amounts to be executed (Y)and arbitrary amounts of opening orders (K). The equilibrium is uniquesince a unique total premium investment, T, is associated with a uniquevector of equilibrium prices, p, by the solution of the eigensystem ofEquation 7.4.7.

7.5 Sell Orders

In this embodiment, “sell” orders in a DBAR digital options market orauction are processed as complementary buy orders with limit pricesequal to one minus the limit price of the “sell” order. For example, forthe MSFT Digital Options auction of Section 6, a sell order for the 50calls with a limit price of 0.44 would be processed as a complementarybuy order for the 50 puts (which are complementary to the 50 calls inthe sense that the defined states which are spanned by the 50 puts arethose which are not spanned by the 50 calls) with limit price equal to0.56 (i.e., 1-0.44). In this manner, buy and sell orders, in thisembodiment of this Section 7, may both be entered in terms of notionalpayouts. Selling a DBAR digital call, put or spread for a given limitprice of an order j (w_(j)) is equivalent to buying the complementarydigital call, put, or spread at the complementary limit price of order j(1−w_(j)).

7.6 Arbitrary Payout Options

In this embodiment, a trader may desire an option that has a payoutshould the option expire in the money that varies depending upon whichdefined in-the-money state occurs. For example, a trader may desiretwice the payout if the state [40,50) occurs than if the state [30,40)occurs. Similarly, a trader may desire that an option have a payout thatis linearly increasing over the defined range of in-the-money states(“strips” as defined in Section 6 above) in order to approximate thetypes of options available in non-DBAR, traditional markets. Optionswith arbitrary payout profiles can readily be accommodated with the DBARmethods of the present invention. In particular, the B matrix, asdescribed in Section 7.2 above, can readily represent such options inthis embodiment. For example, consider a DBAR contract with 5 definedstates.

If a trader desires an option that has the payout profile (0,0, 1, 2,3), i.e., an option that is in-the-money only if the last 3 statesoccur, and for which the fourth state has a payout twice the third, andthe fifth state a payout three times the third, then the row of the Bmatrix corresponding to this order is equal to (0,0, 1, 2, 3). Bycontrast, a digital option for which the same three states arein-the-money would have a corresponding entry in the B matrix of(0,0,1,1,1). Additionally, for digital options all prices, bothequilibrium market prices and limit prices, are bound between 0 and 1.This is because all options are equally weighted linear combinations ofthe defined state probabilities. If, however, options with arbitrarypayout distributions are processed, then the linear combinations (asbased upon the rows of the B matrix) will not be weighted equally andprices need not be bounded between 0 and 1. For ease of exposition, thebulk of the disclosure in this Section 7 has assumed that digitaloptions (i.e., equally weighted payouts) are the only options underconsideration.

7.7 Limit Order Book Optimization

In this embodiment of a DBAR digital options exchange or market orauction as described in this Section 7, traders may enter orders fordigital calls, puts, and spreads by placing conditional investment orlimit orders. As indicated previously in Section 6.8, a limit order isan order to buy or sell a digital call, put or spread that contains aprice (the “limit price”) worse than which the trader desires not tohave his order executed. For example, for a buy order of a digital call,put, or spread, a limit order will contain a limit price which indicatesthat execution should occur only if the final equilibrium price of thedigital call, put or spread is at or below the limit price for theorder. Likewise, a limit sell order for a digital option will contain alimit price which indicates that the order is to be executed if thefinal equilibrium price is at or higher than the limit sell price. Allorders are processed as buy orders and are subject to execution wheneverthe order's limit price is greater than or equal to the then prevailingequilibrium price, because sell orders may be represented as buy orders,as described in the previous section.

In this embodiment, accepting limit orders for a DBAR digital optionsexchange uses the solution of a nonlinear optimization problem (oneexample of an OPF). The problem seeks to maximize the sum total ofnotional payouts of orders that can be executed in equilibrium subjectto each order's limit price and the DBAR digital options equilibriumEquation 7.4.7. Mathematically, the nonlinear optimization thatrepresents the DBAR digital options market or auction limit order bookmay be expressed as follows:

$\begin{matrix}{{x^{*} = {\underset{x}{argmax}{\sum\limits_{j = 1}^{n}x_{j}}}}{{{subject}\mspace{14mu} {{to}(1)}\mspace{14mu} {g_{j}(x)}} = {{x_{j}\left( {{\pi_{j}(x)} - w_{j}^{a}} \right)} \leq 0}}{{{(2)\mspace{14mu} 0} \leq x_{j} \leq {{r_{j}(3)}\mspace{14mu} {Hp}}} = {Tp}}} & {7.7{.1}}\end{matrix}$

The objective function of the optimization problem in 7.7.1 is the sumof the payout amounts for all of the limit orders that may be executedin equilibrium. The first constraint, 7.7.1(1), requires that the limitprice be greater than or equal to the equilibrium price for any payoutto be executed in equilibrium (recalling that all orders, including“sell” orders, may be processed as buy orders). The second constraint,7.7.1(2), requires that the execution payout for the order be positiveand less than or equal to the requested payout of the order. The thirdconstraint, 7.7.1(3) is the DBAR digital option equilibrium equation asdescribed in Equation 7.4.7. These constraints also apply to DBAR ordemand-based markets or auctions, in which contingent claims, such asderivatives strategies, are replicated with replicating claims (e.g.,digital options and/or vanilla options), and then evaluated based on ademand-based valuation of these replicating claims, as described inSections 10, 11 and 13 below.

7.8 Transaction Fees

In this embodiment, before solving the nonlinear optimization problem,the limit order prices for “sell” orders provided by the trader areconverted into buy orders (as discussed above) and both buy and “sell”limit order prices are adjusted with the exchange fee or transactionfee, f_(j). The transaction fee can be set for zero, or it can beexpressed as a flat fee as set forth in this embodiment which is addedto the limit order price received for “sell” orders, and subtracted fromthe limit order price paid for buy orders to arrive at an adjusted limitorder price w_(j) ^(a) for order j, as follows:

For a “sell” order j, w _(j) ^(a)=1−w _(j) −f _(j)  7.8.1

For a buy order j, w _(j) ^(a) =a _(j) −f _(j)  7.8.2

Alternatively, if the transaction fee f_(j) is variable, and expressedas a percentage of the limit order price, w_(j), then the limit orderprice may be adjusted as follows:

For a “sell” order j,

w _(j) ^(a)=(1−w _(j))*(1−f _(j))  7.8.3

For a buy order j,

w _(j) ^(a) =w _(j)*(1−f _(j))  7.8.4

The transaction fee f_(j) can also depend on the time of trade, toprovide incentives for traders to trade early or to trade certainstrikes, or otherwise reflect liquidity conditions in the contract.Regardless of the type of transaction fee f_(j), the limit order pricesw_(j) should be adjusted to w_(j) ^(a) before beginning solution of thenonlinear optimization program. Adjusting the limit order price adjuststhe location of the outer boundary for optimization set by the limitingequation 7.7.1(1). After the optimization solution has been reached, theequilibrium prices for each executed order j, π_(j)(x) can be adjustedby adding the transaction fee to the equilibrium price to produce themarket offer price, and by subtracting the transaction fee from theequilibrium price to produce the market bid price. The limit andequilibrium prices for each executed customer order, in an exampleembodiment in which derivative strategies are replicated into a digitalor vanilla replicating basis, and then subject to a demand-basedvaluation, as more fully set forth in Sections 10, 11 and 13, cansimilarly be adjusted with transaction fees.

7.9 An Embodiment of the Algorithm to Solve the Limit Order BookOptimization

In this embodiment, the solution of Equation 7.7.1 can be achieved witha stepping iterative algorithm, as described in the following steps:

-   -   (1) Place Opening Orders: For each state, premium equal to        k_(i), for i=1, 2, . . . , m, is invested. These investments are        called the “opening orders.” The size of such investments, in        this embodiment, are generally small relative to the subsequent        orders.    -   (2) Convert all “sale” orders to complementary buy orders. As        indicated previously in Section 6.8, this is achieved by (i)        identifying the range of defined states i complementary to the        states being “sold”; and (ii) adjusting the limit “price”        (w_(j)) to one minus the original limit “price” (1−w_(j)). Note        that by contrast to the method disclosed in Section 6.8, there        is no need to convert the amount being sold into an equivalent        amount being bought. In this embodiment in this section, both        buy and “sell” orders are expressed in terms of payout (or        notional payout) terms.    -   (3) For all limit orders, adjust the limit “prices” (w_(j),        1−w_(j)) with transaction fee, by subtracting the transaction        fee f_(j): For a “sell” order j, the adjusted limit price w_(j)        ^(a) therefore equals (1−w_(j)+f_(j)), while for a buy order j,        the adjusted limit price w_(j) ^(a) equals (w_(j)−f_(j)).    -   (4) As indicated above in Section 6.8, group the limit orders by        placing all of the limit orders that span or comprise the same        range of defined states into the same group. Sort each group        from the best (highest “price” buy) to the worst (lowest “price”        buy).    -   (5) Establish an initial iteration step size, α_(j)(1). In this        embodiment the initial iteration step size α_(j)(1) may be        chosen to bear some reasonable relationship to the expected        order sizes to be encountered in the DBAR digital options market        or auction. In most applications, an initial iteration step size        α_(j)(1) equal to 100 is adequate. The current step size        α_(j)(κ) will initially equal the initial iteration step size        (α_(j)(κ)=α_(j)(1) for first iteration) until and unless the        current step size is adjusted to a different step size.    -   (6) Calculate the equilibrium to obtain the total investment        amount T and the state probabilities, p, using equation 7.4.7.        Although the eigenvalues can be computed directly, this        embodiment finds T by Newton-Raphson solution of Equation        7.4.1(b). The solution to T and equation 7.4.1(a) is used to        find the p's.    -   (7) Compute the equilibrium order prices π(x) using the p's        obtained in step (5). The equilibrium order prices π(x) are        equal to B*p.    -   (8) Increment the orders (x_(j)) that have adjusted limit prices        (w_(j) ^(a)) greater than or equal to the current equilibrium        price for that order π_(j)(x) (obtained in step (6)) by the        current step size α_(j)(κ), but not to exceed the requested        notional payout of the order, r_(j). Decrement the orders        (x_(j)) that have a positive executed order amount (x_(j)>0) and        have limit prices less than the current equilibrium market price        π_(j)(x) by the current step size α_(j)(κ), but not to an amount        less than zero.    -   (9) Repeat steps (5) to (7) in subsequent iterations until the        values obtained for the executed order amounts (x_(j)'s) achieve        a desired convergence, as measured by certain convergence        criteria (set forth in Step (8)a), periodically adjusting the        current step size α_(j)(κ) and/or the iteration process after        the initial iteration to further progress the stepping iterative        process towards the desired convergence. The adjustments are set        forth in steps (8)b to (8)d.    -   (8)a In this embodiment, the stepping iterative algorithm is        considered converged based upon a number of convergence        criteria. One such criterion is a convergence of the state        probabilities (“prices”) of the individual defined states. A        sampling window can be chosen, similar to the method by which        the rate of progress statistic is measured (described below), in        order to measure whether the state probabilities are fluctuating        or are merely undergoing slight oscillations (say at the level        of 10⁻⁵) that would indicate a tolerable level of convergence.        Another convergence criterion, in this embodiment, would be to        apply a similar rate of progress statistic to the order steps        themselves. Specifically, the iterative stepping algorithm may        be considered converged when all of the rate of progress        statistics in Equation 7.9.1(c) below are tolerably close to        zero. As another convergence criterion, in this embodiment, the        iterative stepping algorithm will be considered converged when,        in possible combination with other convergence criteria, the        amount of payouts to be paid should any given defined state        occur does not exceed the total amount of investment in the        defined states, T, by a tolerably small amount, such as 10⁻⁵*T.    -   (8)b In this embodiment, the step size may be increased and        decreased dynamically based upon the experienced progress of the        iterative scheme. If, for example, the iterative increments and        decrements are making steady linear progress, then it may be        advantageous to increase the step size. Conversely, if the        iterative increments and decrements (“stepping”) is making less        than linear progress or, in the extreme case, is making little        or no progress, then it is advantageous to reduce the size of        the iterative step.        -   In this embodiment, the step size may be accelerated and            decelerated using the following:

$\begin{matrix}{7.9{.1}} & \; \\{\omega = {\mu*\theta}} & (a) \\{{{{mod}\left( \frac{\kappa}{\omega} \right)} = 0},\mspace{14mu} {\kappa > \omega}} & (b) \\{{\gamma_{j}(\kappa)} = \frac{{{x_{j}(\kappa)} - {x_{j}\left( {\kappa - \omega} \right)}}}{\sum\limits_{i = 1}^{\omega}\; {{{x_{j}(i)} - {x_{j}\left( {i - 1} \right)}}}}} & (c) \\{{\alpha_{j}(\kappa)} = \left\{ \begin{matrix}{{\theta^{(\frac{{{\gamma_{j}{(\kappa)}}*\theta} - 1}{\theta})}*{\alpha_{j}\left( {\kappa - 1} \right)}},} & {{\gamma_{j}(\kappa)} > \frac{1}{\theta}} \\{{\theta^{{{\gamma_{j}{(\kappa)}}*\theta} - 1}*{\alpha_{j}\left( {\kappa - 1} \right)}},} & {{\gamma_{j}(\kappa)} \leq \frac{1}{\theta}}\end{matrix} \right.} & (d)\end{matrix}$

-   -   -   where Equation 7.9.1(a) contains the parameters of the            acceleration/deceleration rules. These parameters have the            following interpretation:        -   θ: a parameter that controls the rate of step size            acceleration and deceleration. Typically, the values for            this parameter will range between 2 and 4, indicating that a            maximum range of acceleration from 100-300%.        -   μ: a multiplier parameter, which, when used to multiply the            parameter θ, yields a number of iterations over which the            step size remains unchanged. Typically, the range of values            for this parameter are 3 to 10.        -   ω: the window length parameter, which is the product of θ            and μ over which the step size remains unchanged. The window            parameter is a number of iterations over which the orders            are stepped with a fixed step size. After these number of            iterations, the progress is assessed, and the step size for            each order may be accelerated or decelerated. Based upon the            above described ranges for θ and μ, the range of values for            ω is between 6 and 40, i.e., every 6 to 40 iterations the            step size is evaluated for possible acceleration or            deceleration.        -   κ: the variable denoting the current iteration of the step            algorithm where κ is an integer multiple of the window            length, ω.        -   γ_(j)(κ): a calculated statistic, calculated at every κ^(th)            iteration for each order j. The statistic is a ratio of two            quantities. The numerator is the absolute value of the            difference between the quantity of order j filled at the            iteration corresponding to the beginning of the window and            at the iteration at the end of window. It represents, for            each order j, the total amount of progress made, in terms of            the execution of order j by either incrementing or            decrementing the executed quantity of order j, from the            start of the window to the end of the window iteration. The            denominator is the sum of the absolute changes of the order            execution for each iteration of the window. Thus, if an            order has made no progress, the γ_(j)(κ) statistic will be            zero. If each step has resulted in progress in the same            direction the γ_(j)(κ) statistic will equal one. Thus, in            this embodiment, the γ_(j)(κ) statistic represents the            amount of progress that has been made over the previous            iteration window, with zero corresponding to no progress for            order j and one corresponding to linear progress for order            j.        -   α_(j)(κ): this parameter is the current step size for order            j at iteration count κ. At every κ^(th) iteration, it is            updated using the equation 7.9.1(d). If the γ_(j)(κ)            statistic reflects sufficient progress over the previous            window by exceeding the quantity 1/θ, then 7.9.1(d) provides            for an increase in the step size, which is accomplished            through a multiplication of the current step size by a            number exceeding one as governed by the formula in 7.9.1(d).            Similarly, if the γ_(j)(κ) statistic reflects insufficient            progress by being equal or less than 1/θ, the step size            parameter will remain the same or will be reduced according            to the formula in 7.9.1(d).        -   These parameters are selected, in this embodiment, based            upon, in part, the overall performance of the rules with            respect to test data. Typically, θ=2-4., μ=3-10 and            therefore ω=6-40. Different parameters may be selected            depending upon the overall performance of the rules.            Equation 7.9.1(b) states that the acceleration or            deceleration of an iterative step for each order's executed            amount is to be performed only on the ω-th iteration, i.e.,            o is a sampling window of a number of iterations (say 6-40)            over which the iterative stepping procedure is evaluated to            determine its rate of progress. Equation 7.9.1(c) is the            rate of progress statistic that is calculated over the            length of each sampling window. The statistic is calculated            for each order j on every ω-th iteration and measures the            rate of progress over the previous ω iterations of stepping.            For each order, the numerator is the absolute value of how            much each order j has been stepped over the sampling window.            The larger the numerator, the larger the amount of total            progress that has been made over the window. The denominator            is the sum of the absolute values of the progress made over            each individual step within the window, summed over the            number of steps, ω, in the window. The denominator will be            the same value, for example, whether 10 positive steps of            100 have been made or whether 5 positive steps of 100 and 5            negative steps of 100 have been made for a given order. The            ratio of the numerator and denominator of Equation 7.9.1(c)            is therefore a statistic that resides on the interval            between 0 and 1, inclusive. If, for example, an order j has            not made any progress over the window period, then the            numerator is zero and the statistic is zero. If, however, an            order j has made maximum progress over the window period,            the rate of progress statistic will be equal to 1. Equation            7.9.1(d) describes the rule based upon the rate of progress            statistic. For each order j at iteration κ (where κ is a            multiple of the window length), if the rate of progress            statistic exceeds 1/θ, then the step size is accelerated. A            higher choice of the parameter θ will result in more            frequent and larger accelerations. If the rate of progress            statistic is less than or equal to 1/θ, then the step size            is either kept the same or decelerated. It may be possible            to employ similar and related acceleration and deceleration            rules, which may have a somewhat different mathematical            parameterization as that described above, to the iterative            stepping of the order amount executions.

    -   (8)c In this embodiment, a linear program may be used, in        conjunction with the iterative stepping algorithm described        above, to further accelerate the rate of progress. The linear        program would be employed primarily at the point when a        tolerable level of convergence in the defined state        probabilities has been achieved. When the defined state        probabilities have reached a tolerable level of convergence, the        nonlinear program of Equation 7.7.1 is transformed, with prices        held constant, into a linear program. The linear program may be        solved using widely available techniques and software code. The        linear program may be solved using a variety of numerical        tolerances on the set of linear constraints. The linear program        will yield a result that is either feasible or infeasible. The        result contains the maximum sum of the executed order amounts        (sum of the x_(j)), subject to the price, bounds, and        equilibrium constraints of Equation 7.7.1, but with the prices        (the vector p) held constant. In frequent cases, the linear        program will result in executed order amounts that are larger        than those in possession at the current iteration of the        stepping procedure. After the linear program is solved, the        iterative stepping procedure is resumed with the executed order        amounts from the linear program. The linear program is an        optimization program of Equation 7.7.1 but with the vector p        from the current iteration κ held constant. With prices        constant, constraints (1) and (3) of nonlinear optimization        problem 7.7.1 become linear and therefore Equation 7.7.1 is        transformed from a nonlinear optimization program to a linear        program.

    -   (8)d Once a tolerable level of convergence has been achieved for        the notional payout executed for each order, x_(j), the entire        stepping iterative algorithm to solve Equation 7.7.1 may then be        repeated with a substantially smaller step size, e.g., a step        size, α_(j)(κ), equal to 1 until a higher level of convergence        has been achieved.        This incremental iteration process also applies to determine the        equilibrium prices of the replicating claims in the auction and        the equilibrium prices of the derivatives strategies, and the        premiums of the customer orders, and resolve the set of        equilibrium conditions, as more fully set forth in Sections 10,        11 and 13.

7.10 Limit Order Book Display

In this embodiment of a DBAR digital options market or auction, it maybe desirable to inform market or auction participants of the amount ofpayout that could be executed at any given limit price for any givenDBAR digital call, put, or spread, as described previously in Section6.9. The information may be displayed in such a manner so as to informtraders and other market participants the amount of an order that may bebought and “sold” above and below the current market price,respectively, for any digital call, put, or spread option. In thisembodiment, such a display of information of the limit order bookappears in a manner similar to the data displayed in the followingtable.

TABLE 7.10.1 Current Pricing Strike Spread To Bid Offer Payout Volume<50 0.2900 0.3020 3.3780 110,000,000 <50 PUT Offer Offer Side Volume0.35 140,002,581 0.32 131,186,810 0.31 130,000,410 MARKET PRICE 0.29000.3020 MARKET PRICE 120,009,731 0.28 120,014,128 0.27 120,058,530 0.24Bid Side Volume BidIn Table 7.10.1, the amount of payout that a trader could execute werehe willing to place an order at varying limit prices above the market(for buy orders) and below the market (for “sell” orders) is displayed.As displayed in the table, the data pertains to a put option, say forMSFT stock as in Section 6, at a strike price of 50. The current priceis 0.2900/0.3020 indicating that the last “sale” order could have beenprocessed at 0.2900 (the current bid price) and that the last buy ordercould have been processed at 0.3020 (the current offer price). Thecurrent amount of executed notional volume for the 50 put is equal to110,000,000. The data indicate that a trader willing to place a buyorder with limit price equal to 0.31 would be able to executeapproximately 130,000,000 notional payout. Similarly, a trader willingto place a “sell” order with limit price equal to 0.28 would be able toachieve indicative execution of approximately 120,000,000 in notional.

7.11 Unique Price Equilibrium Proof

The following is a proof that a solution to Equation 7.7.1 results in aunique price equilibrium. The first-order optimality conditions forEquation 5 yield the following complementary conditions:

(1)g _(j)(x)<0→x _(j) =r _(j)

(2)g _(j)(x)>0→x _(j)=0

(3)g _(j)(x)=0→0≦x _(j) ≦r _(j)  7.11.1A

The first condition is that if an order's limit price is higher than themarket price (g_(j)(x)<0), then that order is fully filled (i.e., filledin the amount of the order request, r_(j)). The second condition is thatan order not be filled if the order's limit price is less than themarket equilibrium price (i.e., g_(j)(x)>0). Condition 3 allows fororders to be filled in all or part in the case where the order's limitprice exactly equals the market equilibrium price.

To prove the existence and convergence to a unique price equilibrium,consider the following iterative mapping:

F(x)=x−β*g(x)  7.11.2A

Equation 7.11.2A can be proved to be contraction mapping which for astep size independent of x will globally converge to a uniqueequilibrium, i.e., it can be proven that Equation 2A has a unique fixedpoint of the form

F(x*)=x*  7.11.3A

To first show that F(x) is a contraction mapping, matrix differentiationof Equation 2A yields:

$\begin{matrix}{{{\frac{{F(x)}}{x} = {I - {\beta*{D(x)}}}}{where}{D(x)} = {B*A*Z^{- 1}*B^{T}}}{A_{ij} = \left\{ {{\begin{matrix}{{p_{i}*\left( {1 - p_{i}} \right)},} & {i = j} \\{{{- p_{i}}*p_{j}},} & {i \neq j}\end{matrix}Z_{i,j}} = \left\{ \begin{matrix}{{T - y_{i} + {p_{i}*y_{i}}},} & {i = j} \\{{p_{j}*y_{i}},} & {i \neq j}\end{matrix} \right.} \right.}} & {7.11{.4}A}\end{matrix}$

The matrix D(x) of Equation 4A is the matrix of order price firstderivatives (i.e., the order price Jacobian). Equation 7.11.2A can beshown to be a contraction if the following condition holds:

$\begin{matrix}{{\frac{{F(x)}}{x}} < 1} & {7.11{.5}A}\end{matrix}$

which is the case if the following condition holds:

β*ρ(D)<1,

where

ρ(D)=max(λ_(i)(D)), i.e., the spectral radius of D  7.11.6A

By the Gerschgorin's Circle Theorem the eigenvalues of A are boundedbetween 0 and 1. The matrix Z⁻¹ is a diagonally dominant matrix, allrows of which sum to 1/T. Because of the diagonal dominance, the othereigenvalues of Z⁻¹ are clustered around the diagonal elements of thematrix, and are approximately equal to p_(i)/k_(i). The largesteigenvalue of Z⁻¹ is therefore bounded above by 1/k_(i). The spectralradius of D is therefore bounded between 0 and linear combinations of1/k_(i) as follows:

$\begin{matrix}{{{\rho (D)} \leq L}{L = \frac{1}{\sum\limits_{i = 1}^{m}\frac{1}{k_{i}}}}} & {7.11{.7}A}\end{matrix}$

where the quantity L, a function of the opening order amounts, can beinterpreted as the “liquidity capacitance” of the demand-based tradingequilibrium (mathematically L is quite similar to the total capacitanceof capacitors in series). The function F(x) of Equation 2A is thereforea contraction if

β<L  7.11.8A

Equation 7.11.8A states that a contraction to the unique priceequilibrium can be guaranteed for contraction step sizes no larger thanL, which is an increasing function of the opening orders in thedemand-based market or auction.

The fixed point iteration of Equation 2A converges to x*. Sincey*=B^(T)x*, y* can be used in Equation 7.4.7 to compute the fundamentalstate prices p* and the total quantity of premium invested T*. If thereare linear dependencies in the B matrix, it may be possible to preservep* through a different allocation of the x's corresponding to thelinearly dependent rows of B. For example, consider two orders, x₁ andx₂, which span the same states and have the same limit order price.Assume that r₁=100 and r₂=100 and that x₁*=x₂*=50 from the fixed pointiteration. Then clearly, x₁=100 and x₂=0 may be set without disturbingp*. For example, different order priority rules may give executionprecedence to the earlier submitted identical order. In any event, thefixed point iteration results in a unique price equilibrium, that is,unique in p.

8. NETWORK IMPLEMENTATION

A network implementation of the embodiment described in Section 7 is ameans to run a complete, market-neutral, self-hedging open book of limitorders for digital options. The network implementation is formed from acombination of demand-based trading core algorithms with an electronicinterface and a demand-based limit order book. This embodiment enablesthe exchange or sponsor to create products, e.g., a series ofdemand-based auctions or markets specific to an underlying event, inresponse to customer demand by using the network implementation toconduct the digital options markets or auctions. These digital options,in turn, form the foundation for a variety of investment, riskmanagement and speculative strategies that can be used by marketparticipants. As shown in FIG. 22, whether accessed using secure,browser-based interfaces over web sites on the Internet or an extensionof a private network, the network implementation provides market makerswith all the functionality conduct a successful market or auctionincluding, for example:

-   (1) Order entry. Orders are taken by a market maker's sales force    and entered into the network implementation.-   (2) Limit order book. All limit orders are displayed.-   (3) Indicative pricing and volumes. While an auction or market is in    progress, prices and order volumes are displayed and updated in real    time.-   (4) Price publication. Prices may be published using the market    maker's intranet (for a private network implementation) or Internet    web site (for an Internet implementation) in addition to market data    services such as Reuters and Bloomberg.-   (5) Complete real-time distribution of market expectations. The    network implementation provides market participants with a display    of the complete distribution of expected returns, at all times.-   (6) Final pricing and order amounts. At the conclusion of a market    or an auction, final prices and filled orders are displayed and    delivered to the market maker for entry or export to existing    clearing and settlement systems.-   (7) Auction or Market administration. The network implementation    provides all functions necessary to administer the market or    auction, including start and stop functions, and details and summary    of all orders by customer and salesperson.

A practical example of a demand-based market or auction conducted usingthe network implementation follows. The example assumes that aninvestment bank receives inquiries for derivatives whose payouts arebased upon a corporation's quarterly earnings release. At present, nounderlying tradable supply of quarterly corporate earnings exists andfew investment banks would choose to coordinate the “other side” of sucha transaction in a continuous market.

Establishing the Market or Auction: First, the sponsor of the market orauction establishes and communicates the details that define the marketor auction, including the following:

-   -   An underlying event, e.g., the scheduled release of an earnings        announcement    -   An auction period or trading period, e.g., the specified date        and time period for the market or auction    -   Digital options strike prices, e.g., the specified increments        for each strike        Accepting and Processing Customer Limit Orders: During the        auction or trading period, customers may place buy and sell        limit orders for any of the calls or puts, as defined in the        market or auction details establishing the market or auction.        Indicative and Final Clearing of the Limit Order Book: During        the auction or trading period, the network implementation        displays indicative clearing prices and quantities, i.e., those        that would exist if the order book were cleared at that moment.        The network implementation also displays the limit order book        for each option, enabling market participants to assess market        depth and conditions. Clearing prices and quantities are        determined by the available intersection of limit orders as        calculated according to embodiments of the present invention. At        the end of the auction or trading period, a final clearing of        the order book is performed and option prices and filled order        quantities are finalized. Market participants remit and accept        premium for filled orders. This completes a successful market or        auction of digital options on an event with no underlying        tradable supply.        Summary of Demand-Based Market or Auction Benefits: Demand-based        markets or auctions can operate efficiently without the        requirement of a discrete order match between and among buyers        and sellers of derivatives. The mechanics of demand-based        markets or auctions are transparent. Investment, risk management        and speculative demand exists for large classes of economic        events, risks and variables for which no associated tradable        supply exists. Demand-based markets and auctions meet these        demands.

9. STRUCTURED INSTRUMENT TRADING

In another embodiment, clients can offer instruments suitable to broadclasses of investors. In particular, an opportunity exists forparticipation in demand-based markets or auctions by customers who wouldotherwise not participate because they typically avoid leverage andtrading in derivatives contracts. In this embodiment, these customersmay transact using existing financial instruments or other structuredproducts, for example, risk-linked notes and swaps, simultaneously withcustomers transacting using DBAR contingent claims, for example, digitaloptions, in the same demand-based market or auction.

In this embodiment, a set of one or more digital options are created toapproximate one or more parameters of the structured products, e.g., aspread to LIBOR (London Interbank Offered Rate) or a coupon on arisk-linked note or swap, a note notional (also referred to, forexample, as a face amount of the note or par or principal), and/or atrigger level for the note or swap to expire in-the-money. The set ofone or more digital options may be referred to, for example, as anapproximation set. The structured products become DBAR-enabled products,because, once their parameters are approximated, the customer is enabledto trade them alongside other DBAR contingent claims, for example,digital options.

The approximation, a type of mapping from parameters of structuredproducts to parameters of digital options, could be an automaticfunction built into a computer system accepting and processing orders inthe demand-based market or auction. The approximation or mapping permitsor enables non-leveraged customers to interface with the demand-basedmarket or auction, side by side with leverage-oriented customers whotrade digital options. DBAR-enabled notes and swaps, as well as otherDBAR-enabled products, provide non-leveraged customers the ability toenhance returns and achieve investment objectives in new ways, andincrease the overall liquidity and risk pricing efficiency of thedemand-based market or auction by increasing the variety and number ofparticipants in the market or auction.

9.1 Overview: Customer-Oriented DBAR-Enabled Products

Instruments can be offered to fit distinct investment styles, needs, andphilosophies of a variety of customers. In this embodiment, “clienteleeffects” refers to, for example, the factors that would motivatedifferent groups of customers to transact in one type of DBAR-enabledproduct over another. The following classes of customers may havevarying preferences, institutional constraints, and investment and riskmanagement philosophies relevant to the nature and degree ofparticipation in demand-based markets or auctions:

-   -   Hedge Funds    -   Proprietary Traders    -   Derivatives Dealers    -   Portfolio Managers    -   Insurers and Reinsurers    -   Pension Funds

Regulatory, accounting, internal institutional policies, and otherrelated constraints may affect the ability, willingness, and frequencyof participation in leveraged investments in general and derivativesproducts such as options, futures, and swaps in particular. Hedge fundsand proprietary traders, for instance, may actively trade digitaloptions, but may be unlikely to trade in certain structured noteproducts that have identical risks while requiring significant capital.On the other hand, “real money” accounts such as portfolio managers,insurers, and pension funds may actively trade instruments that bearsignificant event risk, but these real money customers may be unlikelyto trade DBAR digital options bearing identical event risks.

For example, according to the prospectus for their total return fund,one particular fixed income manager may invest in fixed incomesecurities for which the return of principal and payment of interest arecontingent upon the non-occurrence of a specific ‘trigger’ event, suchas a hurricane, earthquake, tornado, or other phenomenon (referred to,for example, as ‘event-linked bonds’). These instruments typically pay aspread to LIBOR should losses not exceed a stipulated level.

On the other hand, a fixed-income manager may not trade in an IndustryLoss Warranty market or auction with insurers (discussed above inSection 3), even though the risks transacted in this market or auction,effectively a market or auction for digital options on property risksposed by hurricanes, may be identical to the risks borne in theunderwritten Catastrophe-linked (CAT) securities. Similarly, thefixed-income manager and other fixed income managers may participatewidely in the corporate bond market, but may participate to a lesserextent in the default swap market (convertible into a demand-basedmarket or auction), even though a corporate bond bears similar risks asa default swap bundled with a traditional LIBOR-based note or swap.

The unifying theme to these clientele effects is that the structure andform in which products are offered can impact the degree of customerparticipation in demand-based markets or auctions, especially for realmoney customers which avoid leverage and trade few, if any, options butactively seek fixed-income-like instruments offering significant spreadsto LIBOR for bearing some event-related risk on an active and informedbasis.

This embodiment addresses these “clientele effects” in the risk-bearingmarkets by allowing demand-based markets or auctions to simultaneouslyoffer both digital options and DBAR-enabled products, such as, forexample, risk-linked FRNs (or floating rate notes) and swaps, todifferent customers within the same risk-pricing, allocation, andexecution mechanism. Thus, hedge funds, arbitrageurs, and derivativesdealers can transact in the demand-based market or auction in terms ofdigital options, while real money customers can transact in thedemand-based market or auction in terms of different sets ofinstruments: swaps and notes paying spreads to LIBOR. For both types ofcustomers, the payout is contingent upon an observed outcome of aneconomic event, for example, the level of the economic statistic at therelease date (or e.g., at the end of the observation period).

9.2 Overview: FRNs and Swaps

For FRN and swap customers, according to this embodiment, a nexus ofcounterparties to contingent LIBOR-based cash flows based upon materialrisky events can be created in a demand-based market or auction.Schematically, the cash flows resemble a multiple counterparty versionof standard FRN or swap LIBOR-based cash flows. FIG. 23 illustrates thecash flows for each participant. The underlying properties of DBARmarkets or auctions will still apply (as described below), the offeringof this event-linked FRN is market-neutral and self-hedging. In thisembodiment, as with other embodiments of the present invention, ademand-based market or auction is created, ensuring that the receiversof positive spreads to LIBOR are being funded, and completely offset, bythose out-of-the-money participants who receive par.

In this example, with actual ECI at 0.9%, the participants, each withtrigger levels of 0.7%, 0.8%, or 0.9% are all in-the-money, and willearn LIBOR plus the corresponding spread for those triggers on. Thoseparticipants with trigger levels above 0.9% receive par.

9.3 Parameters: FRNs and Swaps vs. Digital Options

The following information provides an illustration of parameters relatedto a principal-protected Employment Cost Index(ECI)-linked FRN note andswap and ECI-linked digital options:

-   -   End of Trading Period: Oct. 23, 2001    -   End of Observation Period: Oct. 25, 2001    -   Coupon Reset Date: Oct. 25, 2001        -   (also referred to, for example, as the “FRN Fixing Date”)    -   Note Maturity: Jan. 25, 2002        -   (when par amount needs to be repaid)    -   Option payout date: Jan. 25, 2002        -   (when payout of digital option is paid, can be set to be the            same date as Note Maturity or a different date)    -   Trigger Index: Employment Cost Index (“ECI”)        -   (also known as the strike price for an equivalent DBAR            digital option)    -   Principal Protection: Par

TABLE 9.3 Indicative Trigger Levels and Indicative Pricing Spread toLIBOR* ECI Trigger (%) (bps) 0.7 50 0.8 90 0.9 180 1 350 1.1 800 1.21200 *For the purposes of the example, assume mid-market LIBOR execution

In this example, a customer (for example, an FRN holder or a noteholder) places an order for an FRN with $100,000,000 par (also referredto, for example, as the face value of the note or notional or principalof the note), selecting a trigger of 0.9% ECI and a minimum spread of180 bps to LIBOR (180 basis points or 1.80% in =addition to LIBOR)during a trading period. After the end of the trading period, Oct. 23,2001, if the market or auction determines the coupon for the note (e.g.,the spread to LIBOR) equal to 200 bps to LIBOR, and the customer's noteexpires in-the-money at the end of the observation period, Oct. 25,2001, then the customer will receive a return of 200 bps plus LIBOR onpar ($100,000,000) on the note maturity date, Jan. 25, 2002.

Alternatively, if the market or auction fixes the rate on the note orsets the spread to 180 bps to LIBOR, and the customer's noteexpires-in-the money at the end of the observation period, then thecustomer will receive a return of 180 bps plus LIBOR (the selectedminimum spread) on par on the note maturity date. If a 3-month LIBOR isequal to 3.5%, and the spread of 180 bps to LIBOR is also for a 3 monthperiod, and the note expires in-the-money, then the customer receives apayout $101,355,444.00 on Jan. 25, 2002, or:

$\begin{matrix}{{{in}\text{-}{the}\text{-}{money}\mspace{14mu} {payout}} = {{par} + {{par} \times \left( {{LIBOR} + {spread}} \right) \times \frac{daycount}{basis}}}} & {9.3A}\end{matrix}$

An “in-the-money note payout” may be a payout that the customer receivesif the FRN expires in-the-money. Analogously, an “out-of-the-money notepayout” may be a payout that the customer receives if the FRN expiresout-of-the-money. “Daycount” is the number of days between the end ofthe coupon reset date and the note maturity date (in this example, 92days). Basis is the number of days used to approximate a year, often setat 360 days in many financial calculations. The variable,“daycount/basis” is the fraction of a year between the observationperiod and the note maturity date, and is used to adjust the relevantannualized interest rates into effective interest rates for a fractionof a year.

If the note expires out-of-the-money, because the ECI is observed to be0.8%, for example, on Oct. 25, 2001 (the end of the observation period),then the customer receives an out-of-the-money payout of par on Jan. 25,2002, the note maturity date, or:

out-of-the-money note payout=par  9.3B

Alternatively, the FRN could be structured as a swap, in which case theexchange of par does not occur. If the swap is structured to adjust theinterest rates into effective interest rates for the actual amount oftime elapsed between the end of the observation period and the notematurity date, then the customer receives a swap payout of $1,355,444.If the ECI fixes below 0.9% (and the swap is structured to adjust theinterest rates), then the FRN holder loses or pays a swap loss of$894,444 or LIBOR times par (see equation 9.3D). The swap payout andswap loss can be formulated as follows:

$\begin{matrix}{{{swap}\mspace{14mu} {payout}} = {{par} \times \left( {{LIBOR} + {spread}} \right) \times \frac{daycount}{basis}}} & {9.3C} \\{{{swap}\mspace{14mu} {loss}} = {{par} \times {LIBOR} \times \frac{daycount}{basis}}} & {9.3D}\end{matrix}$

As opposed to FRNs and swaps, digital options provide a notional or apayout at a digital payout date, occurring on or after the end of theobservation period (when the outcome of the underlying event has beenobserved). The digital payout date can be set at the same time as thenote maturity date or can occur at some other earlier time, as describedbelow. The digital option customer can specify a desired or requestedpayout, a selected outcome, and a limit on the investment amount forlimit orders (as opposed to market orders, in which the customer doesnot place a limit on the investment amount needed to achieve the desiredor requested payout).

9.4 Mechanics: DBAR-Enabling FRNs and Swaps

In this embodiment, as discussed above, both digital options andrisk-linked FRNs or swaps may be offered in the same demand-based marketor auction. Due to clientele effects, traditional derivatives customersmay follow the market or auction in digital option format, while thereal money customers may participate in the market or auction in an FRNformat. Digital options customers may submit orders, inputting optionnotional (as a desired payout), a strike price (as a selected outcome),and a digital option limit price (as a limit on the investment amount).FRN customers may submit orders, inputting a notional note size or par,a minimum spread to LIBOR, and a trigger level or levels, indicating thelevel (equivalently, a strike price) at or above which the FRN will earnthe market or auction-determined spread to LIBOR or the minimum spreadto LIBOR. An FRN may provide, for example, two trigger levels (or strikeprices) indicating that the FRN will earn a spread should the ECI Indexfall between them at the end of the observation period.

In this embodiment, the inputs for an FRN order (which are some of theparameters associated with an FRN) can be mapped or approximated, forexample, at a built-in interface in a computer system, into desiredpayouts, selected outcomes and limits on the investment amounts for oneor more digital options in an approximation set, so that the FRN ordercan be processed in the same demand-based market or auction along withdirect digital option orders. Specifically, each FRN order in terms of anote notional, a coupon or spread to LIBOR, and trigger level may beapproximated with a LIBOR-bearing note for the notional amount (or anote for notional amount earning an interest rate set at LIBOR), and anembedded approximation set of one or more digital options.

As a result of the mapping or approximation, all orders of contingentclaims (for example, digital option orders and FRN orders) are expressedin the same units or variables. Once all orders are expressed in thesame units or variables, an optimization system, such as that describedabove in Section 7, determines an optimal investment amount and executedpayout per order (if it expires in-the-money) and total amount investedin the demand-based market or auction. Then, at the interface, theparameters of the digital options in the approximation set correspondingto each FRN order are mapped back to parameters of the FRN order. Thecoupon for the FRN (if above the minimum spread to LIBOR specified bythe customer) is determined as a function of the digital options in theapproximation set which are filled and the equilibrium price of thefilled digital options in the approximation set, as determined by theentire demand-based market or auction. Thus, the FRN customer inputscertain FRN parameters, such as the minimum spread to LIBOR and thenotional amount for the note, and the market or auction generates otherFRN parameters for the customer, such as the coupon earned on thenotional of the note if the note expires-in-the money.

The methods described above and in section 9.5 below set forth anexample of the type of mapping that can be applied to the parameters ofa variety of other structured products, to enable the structuredproducts to be traded in a demand-based market or auction alongsideother DBAR contingent claims, including, for example, digital options,thereby increasing the degree and variety of participation, liquidityand pricing efficiency of any demand-based market or auction. Thestructured products include, for example, any existing or futurefinancial products or instruments whose parameters can be approximatedwith the parameters of one or more DBAR contingent claims, for example,digital options. The mapping in this embodiment can be used incombination with and/or applied to the other embodiments of the presentinvention.

9.5 Example: Mapping FRNs into Digital Option Space

The following notation, figures and equations illustrate the mapping ofECI-linked FRNs into digital option space, or approximating theparameters of ECI-linked FRNs into parameters of an approximation set ofone or more digital options, and can be applied to illustrate themapping of ECI-linked swaps into digital option space.

9.5.1 Date and Timing Notation and Formulation

-   t_(s): the premium settlement date for the direct digital option    orders and the FRN orders, set at the same time or some time after    the TED (or the end of the trading or auction period).-   t_(E): the event outcome date or the end of the observation period    (e.g., the date of that the outcome of the event is observed).-   t_(O): the option payout date-   t_(R): the coupon reset date, or the date when interest (spread to    LIBOR, including, for example, spread plus LIBOR) begins to accrue    on the note notional.-   t_(N): the note maturity date, or the date for repayment of the    note.-   f: the fraction of the year from date t_(R) to date t_(N). This    number may depend on the'day-count convention used, e.g., whether    the basis for the year is set at 365 days per year or 360 days per    year. In this example, the basis for the year is set at 360 days,    and f can be formulated as follows:

$\begin{matrix}{f = \frac{{number}\mspace{14mu} {of}\mspace{14mu} {days}\mspace{14mu} {between}\mspace{14mu} t_{R}\mspace{14mu} {and}\mspace{14mu} t_{N}}{360}} & {9.5{.1}A}\end{matrix}$

As shown in FIG. 24, the market or auction in this example is structuredsuch that the note maturity date (t_(N)) occurs on or after the optionpayout date (t_(O)) although, for example, the market or auction can bestructured such that t_(N) occurs before t_(O). Additionally, asillustrated, the option payout date (t_(O)) occurs on or after the endof the observation period (t_(E)), and the end of the observation period(t_(E)) occurs on or after the premium settlement date (t_(S)). Thepremium settlement date (t_(S)), can occur on or after the end of thetrading period for the demand-based market or auction. Further, thedemand-based market or auction in this example is structured such thatthe coupon reset date (t_(R)) occurs after the premium settlement date(t_(S)) and before the note maturity date (t_(N)). However, the couponreset date (also referred to, for example, as the “FRN Fixing Date”)(t_(R)) can occur at any time before the note maturity date (t_(N)), andat any time on or after the end of the trading period or the premiumsettlement date (t_(S)). The coupon reset date (t_(R)), for example, canoccur after the end of the observation period (t_(E)) and/or the optionpayout date (t_(O)). In this example, as shown in FIG. 24, the couponreset date (t_(R)) is set between the end of the observation period(t_(E)) and the option payout date (t_(O)).

Similar to the discussion earlier in this specification in Section 1that the duration of the trading period can be unknown to theparticipants at the time that they place their orders, any of the datesabove can be pre-determined and known by the participants at the outset,or they can be unknown to the participants at the time that they placetheir orders. The end of the trading period, the premium settlement dateor the coupon reset date, for example, can occur at a randomly selectedtime, or could occur depending upon the occurrence of some eventassociated or related to the event of economic significance, or upon thefulfillment of some criterion. For example, for DBAR-enabled FRNs, thecoupon reset date could occur after a certain volume, amount, orfrequency of trading or volatility is reached in a respectivedemand-based market or auction. Alternatively, the coupon reset datecould occur, for example, after an nth catastrophic natural event (e.g.,a fourth hurricane), or after a catastrophic event of a certainmagnitude (e.g., an earthquake of a magnitude of 5.5 or higher on theRichter scale), and the natural or catastrophic event can be related orunrelated to the event of economic significance, in this example, thelevel of the ECI.

9.5.2 Variables and Formulation for Demand-Based Market or Auction

-   E: Event of economic significance, in this example, ECI. The level    of the ECI observed on t_(E). This event is the same event for the    FRN and direct digital option orders, referred to, e.g., as a    “Trigger Level” for the FRN order, and as a “strike price” for the    direct digital option order.-   L: London Interbank Offered Rate (LIBOR) from the date t_(R) to    t_(N), a variable that can be fixed, e.g., at the start of the    trading period.-   m: number of defined states, a natural number. Index letter i, i=1,    2 . . . , m.    -   In the example shown in FIG. 9.2, for example, there can be 7        states depending on the outcome of an economic event: the level        of the ECI on the event observation date.    -   ECI<0.7;    -   0.7≦ECI<0.8;    -   0.8≦ECI<0.9;    -   0.9≦ECI<1.0;    -   1.0≦ECI<1.1;    -   1.1≦ECI<1.2; and    -   ECI≧1.2.-   n_(N): number of FRN orders in a demand-based market or auction, a    non-negative integer. Index letter j_(N), j_(N)=1, 2, . . . , n_(N).-   n_(D): number of direct digital option orders in a demand-based    market or auction, a non-negative integer. Index letter j_(D),    j_(D)=1, 2, . . . n_(D). Direct digital option orders, include, for    example, orders which are placed using digital option parameters.-   n_(AD): number of digital option orders in an approximation set for    a j_(N) FRN order. In this example, this number is known and fixed,    e.g., at the start of the trading period, however as described    below, this number can be determined during the mapping process, a    non-negative integer. Index letter z, z=1, 2, . . . n_(AD).-   n: number of all digital option orders in a demand-based market or    auction, a non-negative integer. Index letter j, j=1, 2, . . . n.    -   The above numbers relate to one another in a single demand-based        market or auction as follows:

$\begin{matrix}{n = {{\sum\limits_{j_{N = 1}}^{n_{N}}{n_{AD}\left( j_{N} \right)}} + n_{D}}} & {9.5{.2}A}\end{matrix}$

-   L: the rate of LIBOR from date t_(R) to date t_(N)-   DF_(O): the discount factor between the premium settlement date and    the option payout date (t_(S) and t_(O)), to account for the time    value of money. DFo can be set using LIBOR (although other interest    rates may be used), and equal to, for example, 1/[1+(L* portion of    year from t_(S) to t_(O))].-   DF_(N): the discount rate between the premium settlement date and    the note maturity date, t_(S) and t_(N). DF_(N) can also be set    using LIBOR (although other interest rates may be used), and equal    to, for example, 1/[1+(L*portion of year from t_(S) to t_(N))].

9.5.3 Variables and Formulations for Each Note j_(N) in Demand-BasedMarket or Auction

-   A: notional or face amount or par of note.-   U: minimum spread to LIBOR (a positive number) specified by customer    for note, if the customer's selected outcome becomes the observed    outcome of the event.    -   Although both buy and sell FRN orders can be processed together        with buy and sell direct digital option orders in the same        demand-based market or auction, this example demonstrates the        mapping for a buy FRN order.-   N_(P): The profit on the note if one or more of the states    corresponding to the selected outcome of the event is identified on    the event outcome date as one or more of the states corresponding to    the observed outcome (e.g., the selected outcome turns out to be the    observed outcome, or the ECI reaching or surpassing the Trigger    Level on the event outcome date), at the coupon rate, c, determined    by this demand-based market or auction.

N _(P) =A×c×ƒ×DF _(N)  9.5.3A

-   N_(L): The loss on the note if none of the states corresponding to    the selected outcome of the event is identified on the event outcome    date as one more of the states corresponding to the observed outcome    (e.g., the selected outcome does not turn out to be the observed    outcome, or the ECI does not reach the Trigger Level on the event    outcome date).

N _(L) =A×L×ƒ×DF _(N)  9.5.3B

-   π: the equilibrium price of each of the digital options in the    approximation set that are filled by the demand-based market or    auction, the equilibrium price being determined by the demand-based    market or auction.    -   All of the digital options in the approximation set can have,        for example, the same payout profile or selected outcome,        matching the selected outcome of the FRN. Therefore, all of the        digital options in one approximation set that are filled by the        demand-based market or auction will have, for example, the same        equilibrium price.

9.5.4 Variables and Formulations for Each Digital Option, z, in theApproximation Set of One or More Digital Options for Each Note, j_(N) ina Demand-Based Market or Auction

-   w_(z): digital option limit price for the z^(th) digital option in    the approximation set. The digital options in the approximation set    can be arranged in descending order by limit price. The first    digital option in the set has the largest limit price. Each    subsequent digital option has a lower limit price, but the limit    price remains a positive number, such that w_(z+1)<w_(z). The number    of digital options in an approximation set can be pre-determined    before the order is placed, as in this example, or can be determined    during the mapping process as discussed below.    -   In this example, the limit price for the first digital option        (z=1) in an approximation set for one FRN order (j_(N)) can be        determined as follows:

w ₁ =DF _(O) *L/(U+L)  9.5.4A

-   -   The limit prices for subsequent digital options can be        established such that the differences between the limit prices        in the approximation set become smaller and eventually approach        zero.

-   r_(z): requested or desired payout or notional for the z^(th)    digital option in the approximation set.

-   c: coupon on the FRN, e.g., the spread to LIBOR on the FRN,    corresponding to the coupon determined after the last digital option    order in the approximation set is filled according to the    methodology discussed, for example, in Sections 6 and 7.

The coupon, c, can be determined, for example, by the following:

$\begin{matrix}{c = {L \times \frac{{DF}_{o} - \pi}{w_{z}}}} & {9.5{.4}B}\end{matrix}$

-   -   where w_(z) is the limit price of the last digital option order        z in the approximation set of an FRN, j_(N), to be filled by the        demand-based market or auction.

9.5.5 Formulations for the First Digital Option, z=1, in theApproximation Set of One or More Digital Options for a Note, j_(N) in aDemand-Based Market or Auction

Assuming that the first digital option in the approximation set is theonly digital option order filled by the demand-based market or auction(e.g., w₂<π≦w₁), then following equation 9.5.4B, then:

$\begin{matrix}{c = {L \times \frac{{DF}_{o} - \pi}{w_{1}}}} & {9.5{.5}A}\end{matrix}$

When the equilibrium price (for each of the filled digital options inthe approximation set) is equal to the limit price for the first digitaloption in the approximation set, π=w₁, the digital option profit is r₁(DF_(O)−w₁) and the digital option loss is r₁w₁. Equating the option'sprofit with the note's profit yields:

r ₁(DF ₀ −w ₁)=A*U*f*DF _(N)  9.5.5B

Next, equating the option's loss with the note's loss yields:

r ₁ w ₁ =A*L*f*DF _(N)  9.5.5C

The ratio of the option's profit to the option's loss is equal to theratio of the note's profit to the note's loss:

$\begin{matrix}{\frac{r_{1}\left( {{DF}_{o} - w_{1}} \right)}{r_{1}w_{1}} = \frac{A \times U \times f \times {DF}_{N}}{A \times L \times f \times {DF}_{N}}} & {9.5{.5}D}\end{matrix}$

Simplifying this equation yields:

$\begin{matrix}{\frac{{DF}_{O} - w_{1}}{w_{1}} = \frac{U}{L}} & {9.5{.5}E} \\{\frac{{DF}_{O}}{w_{1}} = {{\frac{U}{L} + 1} = \frac{L + U}{L}}} & {9.5{.5}F}\end{matrix}$

Solving for w₁ yields:

$\begin{matrix}{w_{1} = {\left( \frac{L}{L + U} \right){DF}_{O}}} & {9.5{.5}G}\end{matrix}$

Solving for r₁ from Equation 9.5.5C yields:

r ₁ =A*L*f*DF _(N) /w ₁  9.5.5H

Substituting equation 9.5.5G for w_(i) into equation 9.5.5H yields thefollowing formulation for the requested payout for the first digitaloption in the approximation set:

$\begin{matrix}{r_{1} = \frac{A \times f \times {DF}_{N} \times \left( {L + U} \right)}{{DF}_{O}}} & {9.5{.5}I}\end{matrix}$

9.5.6 Formulations for the Second Digital Option, z=2, in theApproximation Set of one or more digital options for a note, j_(N), in aDemand-Based Market or Auction

Assuming that the second digital option will be filled in theoptimization system for the entire demand-based market or auction, thecoupon earned on the note will be higher than the minimum spread toLIBOR specified by the customer, e.g., c>U.

As stated above, the profit of the FRN is A*c*f*DF_(N) and the loss ifthe states specified do not occur is A*L*f*DF_(N).

Now, since w₁ is determined as set forth above, and w₂ can be set assome number lower than w₁, assuming that the market or auction fillsboth the first and the second digital options and assuming that theequilibrium price is equal to the limit price for the second digitaloption (π=w₂), the profits for the digital options if they expirein-the-money is equal to (r₁+r₂)*(DF_(O)−w₂), and the option loss isequal to (r₁+r₂)*w₂. Equating the option's profit with the note's profityields:

(r ₁ +r ₂)(DF _(O) −w ₂)=A*c*f*DF _(N)  9.5.6A

Equating the option's loss with the note's loss yields:

(r ₁ +r ₂)w ₂ =A*L*f*DF _(N)  9.5.6B

Solving for r2 yields:

r ₂=(A*L*f*DF _(N))/w ₂ −r ₁  9.5.6C

Assuming that the second digital option is the highest order filled inthe approximation set by the demand-based market or auction, the ratioof the profits and losses of both of the options is approximately equalto the profits and losses of the FRN. This approximate equality is usedto solve for the coupon, c. Simplifying the combination of the aboveequations relating to equating the profits and losses of both options tothe profit and loss of the note, yields the following formulation forthe coupon, c, earned on the note if the note expires in-the-money andw₂>π:

c=L*(DF _(O)−π)/w ₂  9.5.6D

9.5.7 Formulations for the z^(th) Digital Option in the ApproximationSet of One or More Digital Options for a Note, j_(N) in a Demand-BasedMarket or Auction

The above description sets forth formulae involved with the first andsecond digital options in the approximation set. The following can beused to determine the requested payout for the z^(th) digital option inthe approximation set. The following can also be used as thedemand-based market's or auction's determination of a coupon for the FRNif the z^(th) digital option is the last digital option in theapproximation set filled by the demand-based market or auction (forexample, according to the optimization system discussed in Section 7),and if the FRN expires in-the-money.

The order of each digital option in the approximation set is treatedanalogously to a market order (as opposed to a limit order), where theprice of the option, π, is set equal to the limit price for the option,w_(z).

Thus, the requested, payout for each digital option, r_(z), in theapproximation set can be determined according to the following formula:

$\begin{matrix}{r_{z} = {\frac{A \times L \times f \times {DF}_{N}}{w_{z}} - {\sum\limits_{x = 1}^{z - 1}r_{x}}}} & {9.5{.7}A}\end{matrix}$

Note that the determination of the requested payout for each digitaloption, r_(z), is recursively dependent on the payouts for the priordigital options, r₁, r₂, . . . , r_(z−1).

The number of digital option orders, n_(AD), used in an approximationset can be adjusted in the demand-based market or auction. For example,an FRN order could be allocated an initial set number of digital optionorders in the approximation set, and each subsequent digital optionorder could be allocated a descending limit order price as discussedabove. After these initial quantities are established for an FRN, therequested payouts for each subsequent digital option can be determinedaccording to equation 9.5.7A. If the requested payout for the z^(th)digital option in the approximation set approaches sufficiently close tozero, where z<n_(AD), then the z^(th) digital option could be set as thelast digital option needed in the approximation set, n_(AD) would thenequal z. The coupon determined by the demand-based market or auctionbecomes a function of LIBOR, the discount factor between the premiumsettlement date and the option payment date, the equilibrium price, andthe limit price of the last digital option in the approximation set tobe filled by the optimization system for the demand-based market orauction discussed in Section 7:

c=L*(DF _(O)−π)/w _(z)  9.5.7B

where w_(z) is the limit price of the last digital option order in theapproximation set to be filled by the optimization system.

9.5.8 Numerical Example of Implementing Formulations for the z^(th)Digital Option in the Approximation Set of One or More Digital Optionsfor a Note, j_(N) in a Demand-Based Market or Auction

The following provides an illustration of a principal-protectedEmployment Cost Index-linked Floating Rate Note. In this numericalexample, the auction premium settlement date t_(S) is Oct. 24, 2001; theevent outcome date t_(E), the coupon reset date t_(R), and the optionpayout date are all Oct. 25, 2001; and the note maturity date t_(N) isJan. 25, 2002.

In this case, the discount factors can be solved using a LIBOR rate L of3.5% and a basis of Actual number of days/360:

DF_(O)=0.999903

DF_(N)=0.991135

f=0.255556

(There are 92 days of discounting between Oct. 25, 2001 and Jan. 25,2002, which is used for the computation off and DF_(N))

The customer or note holder specifies, in this example, that the FRN isa principal protected FRN, because the principal or par or face amountor notional is paid to the note holder in the event that the FRN expiresout-of-the-money. The customer specifies the trigger level of the ECI as0.9% or higher, and the customer enters an order with a minimum spreadof 150 basis points to LIBOR. This customer will receive LIBOR plus 150bps in arrears on 100 million USD on Jan. 25, 2002, plus par if the ECIindex fixes at 0.9% or higher. This customer will receive 100 millionUSD (since the note is principal protected) on Jan. 25, 2002 if the ECIindex fixes at lower than 0.9%.

Following the notation for the variables and the formulation presentedabove,

A=$100,000,000.00 (referred to as the par, principal, notional, faceamount of the note)U=0.015, i.e. bidder wants to receive a minimum of 150 basis points overLIBOR

The parameters for the first digital option in the approximation set forthe demand-based market or auction are determined as follows by equation9.5.4A:

w ₁=(0.035/[0.035+0.015])*0.999903=0.70

It is reasonable to set w₂, the limit price for the second digitaloption order in the approximation set to be equal to 0.69, therefore byequation 9.5.5H:

r ₁=$100,000,000*0.035*0.255556*0.991135/0.70=$1,266,500

The coupon, c, equals 0.015 or 150 basis points, if the first digitaloption order becomes the only digital option order filled by thedemand-based market or auction and the equilibrium price is equal to thelimit price for the first digital option (π=0.7).

The parameters for the second digital option in the approximation setfor the demand-based market or auction are determined as follows,setting the limit price for this digital option to be less than thelimit price for the first digital option, or w₂=0.69, then by equation9.5.6C:

r ₂=$100,000,000*0.035*0.255556*0.991135/0.69−$1,266,500=$18,306

If π, the equilibrium price of the digital option, is between 0.69 (w₂)and 0.70 (w₁), e.g., π=0.695, then the note coupon,c=0.0152=0.035*(0.999903−0.695)/0.70, or 152 bps spread to LIBOR byequation 9.5.5A. This becomes the coupon for the note if thedemand-based market or auction only fills the first digital option orderin the approximation set and if the demand-based market or auction setsthe equilibrium price for the selected outcome equal to 0.695.

If π, the equilibrium price of the digital option, is equal to 0.69(w₂), the coupon for the FRN becomes 157 basis points if the seconddigital option is the highest digital option order filled by thedemand-based market or auction, by equation 9.5.6D:

c=0.035*(0.999903−0.69)/0.69=0.0157 or 157 basis points

The requested payouts for each subsequent digital option, and thesubsequently determined coupon on the note (determined pursuant to thelimit price of the last digital option in the approximation set to befilled by the demand-based market or auction and the equilibrium pricefor the selected outcome), are determined using equations 9.5.7A and9.5.7B.

9.6 Conclusion

These equations present one example of how to map FRNs and swaps intoapproximation sets comprised of digital options, transforming these FRNsand swaps into DBAR-enabled FRNs and swaps. The mapping can occur at aninterface in a demand-based market or auction, enabling otherwisestructured instruments to be evaluated and traded alongside digitaloptions, for example, in the same optimization solution. As shown inFIG. 25, the methods in this embodiment can be used to createDBAR-enabled products out of any structured instruments, so that avariety of structured instruments and digital options can be traded andevaluated in the same efficient and liquid demand-based market orauction, thus significantly expanding the potential size of demand-basedmarkets or auctions.

10. REPLICATING DERIVATIVES STRATEGIES USING DIGITAL OPTIONS

Financial market participants express market views and construct hedgesusing a number of derivatives strategies including vanilla calls andvanilla puts, combinations of vanilla calls and puts including spreadsand straddles, forward contracts, digital options, and knockout options.This section shows how an entity or auction sponsor running ademand-based or DBAR auction can receive and fill orders for thesederivatives strategies.

These derivatives strategies can be included in a DBAR auction using areplicating approximation, a mapping from parameters of, for example,vanilla options to digital options (also referred to as “digitals”), or,as described further in Section 11, a mapping from parameters of, forexample, derivative strategies to a vanilla replicating basis. Thismapping could be an automatic function built into a computer systemaccepting and processing orders in the demand-based market or auction.The replicating approximation permits or enables auction participants orcustomers to interface with the demand-based market or auction, side byside with customers who trade digital options, notes and swaps, as wellas other DBAR-enabled products. This increases the overall liquidity andrisk pricing efficiency of the demand-based market or auction byincreasing the variety and number of participants in the market orauction. FIG. 26 shows how these options may be included in a DBARauction with a digital replicating basis. FIG. 29 shows how theseoptions may be included in a DBAR auction with a vanilla replicatingbasis.

Offering such derivatives strategies in a DBAR auction provides severalbenefits for the customers. First, customers may have access to two-waymarkets for these derivatives strategies giving customers transparencynot currently available in many derivatives markets. Second, customerswill receive prices for the derivatives strategies based on the pricesof the underlying digital claims, insuring that the prices for thederivatives strategies are fairly determined. Third, a DBAR auction mayprovide customers with greater liquidity than many current derivativesmarkets: in a DBAR auction, customers may receive a lower bid-ask spreadfor a given notional size executed and customers may be able to executemore notional volume for a given limit price. Finally, offering theseoptions provides customers the ability to enhance returns and achieveinvestment objectives in' new ways.

In addition, offering such derivative strategies in a DBAR auctionprovides benefits for the auction sponsor. First, the auction sponsorwill earn fee income from these orders. In addition, the auction sponsorhas no price making requirements in a DBAR auction as prices aredetermined endogenously. In offering these derivatives strategies, theauction sponsor may be exposed to the replication profit and loss orreplication P&L—the risk deriving from synthesizing the variousderivatives strategies using only digital options. However, this riskmay be small in a variety of likely instances, and in certain instancesdescribed in Section 11, when derivative strategies are replicated intoa vanilla replicating basis, this risk may be reduced to zero.Regardless, the cleared book from a DBAR auction, excluding thisreplication P&L and opening orders, will be risk-neutral andself-hedging, a further benefit for the auction sponsor.

The remainder of section 10 shows how a number of derivatives strategiescan be replicated in a DBAR auction. Section 10.1 shows how to replicatea general class of derivatives strategies. Next, section 10.2 appliesthis general result for a variety of derivatives strategies. Section10.3 shows how to replicate digitals using two distributional models forthe underlying. Section 10.4 computes the replication P&L for a set oforders in the auction. Appendix 10A summarizes the notation used in thissection. Appendix 10B derives the mathematics behind the results insection 10.1 and 10.2. Appendix 10C derives the mathematics behindresults in section 10.3.

10.1 The General Approach to Replicating Derivatives Strategies withDigital Options

Let U denote the underlying measurable event and let Ω denote the samplespace for U.

U may be a univariate random variable and thus Ω may be, for example, R¹or R⁺. Otherwise U may be a multidimensional random variable and Ω maybe, for example, R^(n).

Assume that the sample space Ω is divided into S disjoint and non-emptysubsets Ω₁, Ω₂, . . . , Ω_(S) such that

Ω_(i)∩Ω_(j)=Ø1≦i≦S and 1≦j≦D and i≠j  10.1A

Ω₁∪Ω₂∪ . . . Ω_(S)=Ω  10.1B

Thus, Ω₁, Ω₂, . . . , Ω_(S) represents a mutually exclusive andcollectively exhaustive division of Ω.

Each sample space partition Ω_(s) can be associated with a state s.Namely, the underlying UεΩ_(s) that means that state s has occurred, fors=1, 2, . . . , S. Thus, there are S states in totality. It is worthnoting that this definition of “state” differs from other definitions of“state” in that a “state” may represent only a specific outcome of asample space: in this example embodiment, a “state” may represent a setof multiple outcomes.

Denote the probability of state s occurring as p_(s) for s=1, 2, . . . ,S. Thus,

p_(s)≡Pr[U:UεΩ_(s)] for s=1, 2, . . . , S  10.1C

Assume that p_(s)>0 for s=1, 2, . . . , S.

Consider a derivatives strategy that pays out d(U). This derivativesstrategy will be referred to using the function d. The function d may bequite general: d may be a continuous or discontinuous function of U, adifferentiable or non-differentiable function of U. For example, in thecase where a derivatives strategy based on digitals is being replicated,the function d is discontinuous and non-differentiable.

Let a_(s) denote the digital replication for state s, the series ofdigitals that replicate the derivatives strategy d. Let C denote thereplication P&L to the auction sponsor. If C is positive (negative),then the auction sponsor receives a profit (a loss) from the replicationof the strategy. The replication P&L to the auction sponsor C is givenby the following formula for a buy order of d

$\begin{matrix}{C \equiv {\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}\left\lbrack {a_{s} - {d(U)} + \underset{\_}{e}} \right\rbrack}}} & {10.1D}\end{matrix}$

where

$\begin{matrix}{\underset{\_}{e} = {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}{E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}}} & {10.1E}\end{matrix}$

In this case, e denotes the minimum conditional expected value of d(U)within state s. For intuition as to why C depends on e, consider thesimple example where d(U)=ξ (a constant) for all values of U. In thiscase, of course, the replication P&L should be zero since there are nodigitals required to replicate the strategy d. It can be shown that e=ξand C equals 0 for a_(s)=0 for s=1, 2, . . . , S using equation 10.1D.Thus, e is required in equation 10.1D to make C, the replication P&L,zero in this case.

The replication P&L for a sell of d is the negative of the replicationP&L of a buy of d

$\begin{matrix}{C \equiv {\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}\left\lbrack {{d(U)} - a_{s} - \underset{\_}{e}} \right\rbrack}}} & {10.1F}\end{matrix}$

In equation 10.1F, a_(s) represents the replicating digital for a buyorder.

Let

$\begin{matrix}{\overset{\_}{e} = {\max\limits_{{s = 1},2,\ldots \mspace{14mu},S}{E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}}} & {10.1G}\end{matrix}$

As defined in 10.1G, ē denotes the maximum conditional expected value ofd(U) within state s.

This example embodiment restricts these parameters

0≦e<ē≦∞  10.1H

so that the conditional expected value of d is bounded above and below.Note that this condition can be met when the function d itself isunbounded, as is the case for many derivatives strategies such asvanilla calls and vanilla puts.

Values of (a₁, a₂, . . . , a_(S−1), a_(S)) are selected in this exampleembodiment as follows

-   -   Objective: Choose (a₁, a₂, . . . , a_(S−1), a_(S)) to minimize        Var[C] subject to E[C]=0        In words, the a's are selected so that the auction sponsor has        the minimum variance of replication P&L subject to the        constraint that the expected replication P&L is zero.

Under these conditions, the general replication theorem in appendix 10Bproves that the replication digitals are

a _(s) =E[d(U)|UεΩ _(s) ]−e for s=1, 2, . . . , S  10.1I

The replication P&L and the infimum replication P&L can be computed asfollows

$\begin{matrix}{C = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}\left( {{E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack} - {d(U)}} \right)}}} & {10.1J} \\{{\inf \; C} = {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}\left\lbrack {\inf\limits_{U \in \Omega_{s}}\left( {{E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack} - {d(U)}} \right)} \right\rbrack}} & {10.1K}\end{matrix}$

The infimum is significant because it represents the worst possible lossto the auction sponsor. If d is bounded over the sample space, then thisinfimum will be finite, but in the case where d is unbounded thisinfimum may be unbounded below.

For an order to sell the derivatives strategy d, the general replicationtheorem in appendix 10B shows that the replicating digitals for sellingd are

a _(s) =ē−E[d(U)|UεΩ _(s)] for s=1, 2, . . . , S  10.1L

The replication P&L and the infimum replication P&L for a sell of d canbe computed as follows

$\begin{matrix}{C = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}\left( {{d(U)} - {E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}} \right)}}} & {10.1M} \\{{\inf \; C} = {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}\left\lbrack {\inf\limits_{U \in \Omega_{s}}\left( {{d(U)} - {E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}} \right)} \right\rbrack}} & {10.1N}\end{matrix}$

Note that the replication P&L for a sell of d is the negative of thereplication P&L for a buy of d. Similarly, the infimum replication P&Lfor a sell of d is the negative of the infimum replication P&L for a buyof d.

The variance of the replication P&L is the same for a buy or a sell

$\begin{matrix}{{{Var}\lbrack C\rbrack} = {\sum\limits_{s = 1}^{S}{p_{s}{{Var}\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}}}} & {10.1O}\end{matrix}$

It is worth noting that for both buys of d and sells of d that

min(a ₁ , a ₂ , . . . , a _(S−1) , a _(S))=0  10.1P

Thus, all the a's are non-negative and at least one of the a's isexactly zero.

The example embodiment described above restricts the parameters asfollows

0≦e<ē<∞  10.1Q

Equation 10.1Q requires the conditional expected value of d(U) to bebounded both above and below. Other example embodiments may relax thisassumption. For example, values of (a₁, a₂, . . . , a_(S−1), a_(S))could be selected such that

-   -   Objective: Choose (a₁, a₂, . . . , a_(S−1), a_(S)) to minimize        E[median|C|] subject to median[C]=0

In this case, the a's are selected so that the auction sponsor has thelowest average absolute replication P&L subject to the constraint thatthe median replication P&L is zero. This objective function allows asolution for the (a₁, a₂, . . . , a_(S−1), a_(S)) where the conditionalexpected values of d(U) can be unbounded.

In addition to replicating these derivatives strategies, one candetermine pricing on a derivatives strategy based on the replicatingdigitals. In an example embodiment, the price of a derivatives strategyd will be

${{DF} \times {\sum\limits_{s = 1}^{S}{a_{s}p_{s}}}},$

where the a's represent the replicating digitals for the strategy d andDF represents the discount factor (which is based on the funding ratebetween the premium settlement date and the notional settlement date).In the case where the discount factor DF equals 1, the price of aderivatives strategy d will be

$\sum\limits_{s = 1}^{S}{a_{s}{p_{s}.}}$

In an example embodiment, the auction sponsor may assess a fee for acustomer transaction thus increasing the customer's price for a buy anddecreasing the customer's price for a sell. This fee may be based on thereplication P&L associated with each strategy, charging possibly anincreasing amount based on but not limited to the variance ofreplication P&L or the infimum replication P&L for a derivativesstrategy d.

10.2 Application of General Results to Special Cases

This section begins by examining the special case where the underlying Uis one-dimensional. Section 10.2.1 introduces the general result andthen section 10.2.2 provides specific examples for a one-dimensionalunderlying. Section 10.2.3 provides results for a two-dimensionalunderlying and section 10.2.4 provides results for higher dimensions.

10.2.1 General Result

For a one-dimensional underlying U, let the strikes be denoted as k₁,k₂, . . . , k_(S−1). Let the strikes be in increasing order, that is,

k ₁ <k ₂ <k ₃ < . . . <k _(S−2) <k _(S−1)  10.2.1A

For notational purposes, let k₀=−∞ and let k_(S)=+∞. Therefore,

Ω₁=[U:k₀≦U<k₁]=[U:U<k₁]  10.2.1B

Ω_(S) =[U:k _(S−1) ≦U<k _(S) ]=[U:k _(S−1) ≦U]  10.2.1C

and thus Ω=R¹ and

Ω_(s) =[U:k _(s−1) ≦U<k _(s)]s=1, 2, . . . , S  10.2.1C

In other example embodiments Ω=R⁺, which may be useful for example ifthe underlying U represents the price of an instrument that cannot benegative.

The replicating digitals for a buy for a one-dimensional underlying is

a _(s) =E[d(U)|k _(s−1) ≦U<k _(s)]− e for s=1, 2, . . . , S  10.2.1E

where

$\begin{matrix}{\underset{\_}{e} = {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}{E\left\lbrack {{d(U)}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}}} & {10.2{.1}F}\end{matrix}$

The replication P&L and the infimum replication P&L are

$\begin{matrix}{\mspace{20mu} {C = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left( {{E\left\lbrack {{d(U)}{k_{s - 1} \leq U < k_{s}}} \right\rbrack} - {d(U)}} \right)}}}} & {10.2{.1}G} \\{{\inf \; C} = {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}\left\lbrack {\inf\limits_{k_{s - 1} \leq U < k_{s}}\left( {{E\left\lbrack {{d(U)}{k_{s - 1} \leq U < k_{s}}} \right\rbrack} - {d(U)}} \right)} \right\rbrack}} & {10.2{.1}H}\end{matrix}$

For sells of the derivatives strategy d the replicating digitals are

a _(s) =ē−E[d(U)|k _(s−1) ≦U<k _(s)] for s=1, 2, . . . , S  10.2.1I

where

$\begin{matrix}{\overset{\_}{e} = {\max\limits_{{s = 1},2,\ldots \mspace{14mu},S}{E\left\lbrack {{d(U)}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}}} & {10.2{.1}J}\end{matrix}$

Further, the replication P&L and the infimum replication P&L are

$\begin{matrix}{\mspace{20mu} {C = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left( {{d(U)} - {E\left\lbrack {{d(U)}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}} \right)}}}} & {10.2{.1}K} \\{{\inf \; C} = {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}\left\lbrack {\inf\limits_{k_{s - 1} \leq U < k_{s}}\left( {{d(U)} - {E\left\lbrack {{d(U)}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}} \right)} \right\rbrack}} & {10.2{.1}L}\end{matrix}$

The variance of replication P&L for both buys and sells of d is

$\begin{matrix}{{{Var}\lbrack C\rbrack} = {\sum\limits_{s = 1}^{S}{p_{s}{{Var}\left\lbrack {{d(U)}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}}}} & {10.2{.1}M}\end{matrix}$

Section 10.2.2 uses these formulas to derive results for derivativesstrategies on one-dimensional underlyings.

10.2.2 Replicating Derivatives Strategies with a One-DimensionalUnderlying

This section uses the formulas from section 10.2.1 to computereplicating digitals (a₁, a₂, . . . , a_(S−1), a_(S)) for both buys andsells of the following derivatives strategies: digital options (digitalcalls, digital puts, and range binaries), vanilla call options andvanilla put options, call spreads and put spreads, straddles, collaredstraddles, forwards, collared forwards, fixed price digital options, andfixed price vanilla options.

In addition to these derivatives strategies, an auction sponsor canoffer derivatives based on these techniques, including but not limitedto derivatives that are quadratic (or higher power) functions of theunderlying, exponential functions of the underlying, and butterfly orcombination strategies that generally require the buying and selling ofthree of more options.

Replicating Digital Calls, Digital Puts and Range Binaries

A digital call expires in-the-money and pays out a specified amount ifthe underlying U is greater than or equal to a threshold value. Fornotation, let ν be an integer such that 1≦v≦S−1. Then the d function fora digital call with a strike price of k_(v) is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {{{for}\mspace{14mu} U} < k_{v}} \\1 & {{{for}\mspace{14mu} k_{v}} \leq U}\end{matrix} \right.} & {10.2{.2}A}\end{matrix}$

For a buy order of a digital call with a strike price of k_(v) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\1 & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}B}\end{matrix}$

For a sell order of a digital call with a strike price of k_(v) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}1 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\0 & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}C}\end{matrix}$

A digital put pays out a specific quantity if the underlying is strictlybelow a threshold on expiration. Let v be an integer such that 1≦v≦S−1.For a digital put, d is defined as

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}1 & {{{for}\mspace{14mu} U} < k_{v}} \\0 & {{{for}\mspace{14mu} k_{v}} \leq U}\end{matrix} \right.} & {10.2{.2}D}\end{matrix}$

For a buy order of a digital put with a strike price of k_(v) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}1 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\0 & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}E}\end{matrix}$

For a sell order of a digital put with a strike price of k_(v) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\1 & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}F}\end{matrix}$

A range binary strategy pays out a specific amount if the underlying iswithin a specified range. Let v and w be integers such that 1≦v<w≦S−1.Then the range binary strategy can be represented as

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {{{for}\mspace{14mu} U} < k_{v}} \\1 & {{{for}\mspace{14mu} k_{v}} \leq U < k_{w}} \\0 & {{{for}\mspace{14mu} k_{w}} \leq U}\end{matrix} \right.} & {10.2{.2}G}\end{matrix}$

For a buy order of a range binary with strikes k_(v) and k_(w) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\1 & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},w} \\0 & {{{{for}\mspace{14mu} s} = {w + 1}},{w + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}H}\end{matrix}$

For a sell order of a range binary with strikes k_(v) and k_(w) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}1 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\0 & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},w} \\1 & {{{{for}\mspace{14mu} s} = {w + 1}},{w + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}I}\end{matrix}$

For these three digital strategies, it can be shown that e=0 and ē=1.For buys and sells of digital calls, digital puts, and range binaries,the replication P&L is zero and the variance of the replication P&L iszero.

Replicating Vanilla Call Options and Vanilla Put Options

This section describes how to replicate vanilla calls and vanilla puts.Though financial market participants will often just refer to theseoptions as simply calls and puts, the modifier vanilla is used here todifferentiate these calls and puts from digital calls and digital puts.

Let v denote an integer such that 1≦v≦S−1. A vanilla call pays out asfollows

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {{{for}\mspace{14mu} U} < k_{v}} \\{U - k_{v}} & {{{for}\mspace{14mu} k_{v}} \leq U}\end{matrix} \right.} & {10.2{.2}J}\end{matrix}$

For a buy order for a vanilla call with strikes of k_(v) the replicatingdigitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\{{E\left\lbrack {U{k_{s - 1} \leq U < k_{s}}} \right\rbrack} - k_{v}} & {{{{for}\mspace{14mu} s} = {v + 1}},{v = 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}K}\end{matrix}$

For a sell order for a vanilla call with strike k_(v) the replicatingdigitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{E\left\lbrack {U{k_{S - 1} \leq U}} \right\rbrack} - k_{v}} & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\\begin{matrix}{{E\left\lbrack {U{k_{S - 1} \leq U}} \right\rbrack} -} \\{E\left\lbrack {U{k_{s - 1} \leq U < k_{s}}} \right\rbrack}\end{matrix} & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{11mu},S}\end{matrix} \right.} & {10.2{.2}L}\end{matrix}$

Note that for a vanilla call, e=0 and ē=E[U|k_(s−1)≦U]−k_(v).

FIGS. 27A, 27B and 27C show the functions d and C for a vanilla calloption for an example that is discussed in further detail in sections10.3.1 and 10.3.2.

For a vanilla put, let v be an integer such that 1≦v≦S−1. A vanilla putpays out as follows

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{{k_{v} - {U\mspace{14mu} {for}\mspace{14mu} U}} < k_{v}} \\{{0\mspace{14mu} {for}\mspace{14mu} k_{v}} \leq U}\end{matrix} \right.} & {10.2{.2}M}\end{matrix}$

For a buy order for a vanilla put with strikes of k_(v) the replicatingdigitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{k_{v} - {{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s}} = 1},2,\ldots \mspace{11mu},v} \\{{{0\mspace{14mu} {for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{11mu},S}\end{matrix} \right.} & {10.2{.2}N}\end{matrix}$

For a sell order for a vanilla put with a strike of k_(v) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} - {{E\left\lbrack U \middle| {U < k_{1}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s}} = 1},2,\ldots \mspace{11mu},v} \\{{{k_{v} - {{E\left\lbrack U \middle| {U < k_{1}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s}} = {v + 1}},{v + 2},\ldots \mspace{11mu},w}\end{matrix} \right.} & {10.2{.2}O}\end{matrix}$

Note that for a vanilla put, e=0 and ē=k_(v)−E[U|U<k₁].

It is worth noting that the replication P&L for a buy or sell of avanilla call and vanilla put can be unbounded because these options canpay out unbounded amounts.

Replicating Call Spreads and Put Spreads

A buy of a call spread is the simultaneous buy of a vanilla call and thesell of a vanilla call. Let v and w be integers such that 1≦v<w≦S−1.Then d for a call spread is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{{0\mspace{14mu} {for}\mspace{14mu} U} < k_{v}} \\{{U - {k_{v}\mspace{14mu} {for}\mspace{14mu} k_{v}}} \leq U < k_{w}} \\{{k_{w} - {k_{v}\mspace{14mu} {for}\mspace{14mu} k_{w}}} \leq U}\end{matrix} \right.} & {10.2{.2}P}\end{matrix}$

For a buy order for a call spread with strikes of k_(v) and k_(w) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{0\mspace{14mu} {for}\mspace{14mu} s} = 1},2,\ldots \mspace{11mu},v} \\{{{{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} - {k_{v}\mspace{14mu} {for}\mspace{14mu} s}} = {v + 1}},{v + 2},\ldots \mspace{11mu},w}\;} \\{{{k_{w} - {k_{v}\mspace{14mu} {for}\mspace{14mu} s}} = {w + 1}},{w + 2},\ldots \mspace{11mu},S}\end{matrix} \right.} & {10.2{.2}Q}\end{matrix}$

For a sell order for a call spread with strikes of k_(v) and k_(w) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{k_{w} - {k_{v}\mspace{14mu} {for}\mspace{14mu} s}} = 1},2,\ldots \mspace{11mu},v} \\{{{k_{w} - {{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s}} = {v + 1}},{v + 2},\ldots \mspace{11mu},w} \\{{{0\mspace{14mu} {for}\mspace{14mu} s} = {w + 1}},{w + 2},\ldots \mspace{11mu},S}\end{matrix} \right.} & {10.2{.2}R}\end{matrix}$

If strike k_(w) is high enough, a call spread will approximate a vanillacall. However, note that the replication P&L for a call spread is alwaysbounded, whereas the replication P&L for a vanilla call can be infinite.

FIGS. 28A, 28B and 28C show the functions d and C for a call spread foran example that is discussed in further detail in sections 10.3.1 and10.3.2.

A buy of a put spread is the simultaneous buy of a vanilla put and asell of a vanilla put. Let v and w be integers such that 1≦v<w≦S−1. Thenfor a put spread the function d is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{{k_{w} - {k_{v}\mspace{14mu} {for}\mspace{14mu} U}} < k_{v}} \\{{k_{w} - {U\mspace{14mu} {for}\mspace{14mu} k_{v}}} \leq U < k_{w}} \\{{0\mspace{14mu} {for}\mspace{14mu} k_{w}} \leq U}\end{matrix} \right.} & {10.2{.2}S}\end{matrix}$

For a buy order for a put spread with strikes of k_(v) and k_(w) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{k_{w} - {k_{v}\mspace{14mu} {for}\mspace{14mu} s}} = 1},2,\ldots \mspace{11mu},v} \\{{{k_{w} - {{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s}} = {v + 1}},{v + 2},\ldots \mspace{11mu},w} \\{{{0\mspace{14mu} {for}\mspace{14mu} s} = {w + 1}},{w + 2},\ldots \mspace{11mu},S}\end{matrix} \right.} & {10.2{.2}T}\end{matrix}$

For a sell order for a put spread with strikes of k_(v) and k_(w) thereplicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{0\mspace{14mu} {for}\mspace{14mu} s} = 1},2,\ldots \mspace{11mu},v} \\{{{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} - {k_{v}\mspace{14mu} {for}\mspace{14mu} s}} = {v + 1}},{v + 2},\ldots \mspace{11mu},w} \\{{{k_{w} - {k_{v}\mspace{14mu} {for}\mspace{14mu} s}} = {w + 1}},{w + 2},\ldots \mspace{11mu},S}\end{matrix} \right.} & {10.2{.2}U}\end{matrix}$

For a strike k_(v) low enough, this put spread will approximate avanilla put. However, note that the replication P&L for a put spread isalways bounded, whereas the replication P&L for a vanilla put can beinfinite.

For call spreads and put spreads note that e=0 and ē=k_(w)−k_(v).

Replicating Straddles and Collared Straddles

A buy of a straddle is the simultaneous buy of a call and a put bothwith identical strike prices. A buy of a straddle is generally a bullishvolatility strategy, in that the purchaser profits if the outcome isvery low or very high. Using digitals one can construct straddles asfollows.

Let v be an integer such that 2≦v≦S−2. For a straddle, the payout d is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{{k_{v} - {U\mspace{14mu} {for}\mspace{14mu} U}} < k_{v}} \\{{U - {k_{v}\mspace{14mu} {for}\mspace{14mu} k_{v}}} \leq U}\end{matrix} \right.} & {10.2{.2}V}\end{matrix}$

For a buy order of a straddle with strike k_(v) the replicating digitalsare

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{k_{v} - {E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} - {\underset{\_}{e}\mspace{14mu} {for}\mspace{14mu} s}} = 1},2,\ldots \mspace{11mu},v} \\{{{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} - k_{v} - {\underset{\_}{e}\mspace{14mu} {for}\mspace{14mu} s}} = {v + 1}},{v + 2},\ldots \mspace{11mu},S}\end{matrix} \right.} & {10.2{.2}W}\end{matrix}$

where

e =min[k _(v) −E[U|k _(v−1) ≦U<k _(v) ],E[U|k _(v) ≦U<k _(v+1) ]−k_(v)]  10.2.2X

For the sell of a straddle with strike k_(v) the replicating digitalsare

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{\overset{\_}{e} - k_{v} + {{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s}} = 1},2,\ldots \mspace{11mu},v} \\{{{\overset{\_}{e} - {E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} + {k_{v}\mspace{14mu} {for}\mspace{14mu} s}} = {v + 1}},{v + 2},\ldots \mspace{11mu},S}\end{matrix} \right.} & {10.2{.2}Y}\end{matrix}$

where

e =max[k _(v) −E[U|U<k ₁ ],E[U|k _(S−1) ≦U]−k _(v)]  10.2.2Z

Note that, buys and sells of straddles may have unbounded replicationP&L since the underlying vanilla calls and vanilla puts themselves canhave unbounded payouts.

As opposed to offering straddles, an auction sponsor may instead wish tooffer customers a straddle-like instrument with bounded replication P&L,referred to here as a collared straddle. Let v be an integer such that2≦v≦S−2. For a collared straddle let

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{{k_{v} - {k_{1}\mspace{14mu} {for}\mspace{14mu} U}} < k_{1}} \\{{k_{v} - {U\mspace{14mu} {for}\mspace{14mu} k_{1}}} \leq U < k_{v}} \\{{U - {k_{v}\mspace{14mu} {for}\mspace{14mu} k_{v}}} \leq U < k_{S - 1}} \\{{k_{S - 1} - {k_{v}\mspace{14mu} {for}\mspace{14mu} k_{S - 1}}} \leq U}\end{matrix} \right.} & {10.2{.2}{AA}}\end{matrix}$

For a buy order of a collared straddle with strike k_(v) the replicatingdigitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{k_{v} - k_{1} - {\underset{\_}{e}\mspace{14mu} {for}\mspace{14mu} s}} = 1} \\{{{k_{v} - {E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} - {\underset{\_}{e}\mspace{14mu} {for}\mspace{14mu} s}} = 2},3,\ldots \mspace{11mu},v} \\{{{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} - k_{v} - {\underset{\_}{e}\mspace{14mu} {for}\mspace{14mu} s}} = {v + 1}},{v + 2},\ldots \mspace{11mu},{S - 1}} \\{{k_{S - 1} - k_{v} - {\underset{\_}{e}\mspace{14mu} {for}\mspace{14mu} s}} = S}\end{matrix} \right.} & {10.2{.2}{AB}}\end{matrix}$

where e is as before. For a sell order of a collared straddle withstrike k_(v) the replicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{\overset{\_}{e} - k_{v} + {k_{1}\mspace{14mu} {for}\mspace{14mu} s}} = 1} \\{{{\overset{\_}{e} - k_{v} + {{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s}} = 2},3,\ldots \mspace{11mu},v} \\{{{\overset{\_}{e} - {E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} + {k_{v}\mspace{14mu} {for}\mspace{14mu} s}} = {v + 1}},{v + 2},\ldots \mspace{11mu},{S - 1}} \\{{\overset{\_}{e} - k_{S - 1} + {k_{v}\mspace{14mu} {for}\mspace{14mu} s}} = S}\end{matrix} \right.} & {10.2{.2}{AC}}\end{matrix}$

where ē=max[k_(S−1)−k_(v), k_(v)−k₁].

As observed above, the replication P&L for this collared straddle isbounded, since it comprises the buy of a call spread and the buy of aput spread.

Replicating Forwards and Collared Forwards

A forward pays out based on the underlying as follows

d(U)=U  10.2.2AD

Therefore, for a buy order for a forward, the replicating digitals are

a _(s) =E[U|k _(s−1) ≦U<k _(s) ]−E[U|U<k ₁] for s=1, 2, . . . ,S  10.2.2AE

For a sell order for a forward, note the replicating digitals are

a _(s) =E[U|k _(S−1) ≦U]−E[U|k _(s−1) ≦U<k _(s)] for s=1, 2, . . . ,S  10.2.2AF

Note that for a forward, e=E[U|U<k₁] and ē=E[U|k_(S−1)≦U].

Note that buys and sells of forwards can have unbounded replication P&L.

To avoid offering a forward with possibly unbounded replication P&L, theauction sponsor may offer a collared forward strategy with maximum andminimum payouts. For a collared forward

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{{k_{1}\mspace{14mu} {for}\mspace{14mu} U} < k_{1}} \\{{U\mspace{14mu} {for}\mspace{14mu} k_{1}} \leq U < k_{S - 1}} \\{{k_{S - 1}\mspace{14mu} {for}\mspace{14mu} k_{S - 1}} \leq U}\end{matrix} \right.} & {10.2{.2}{AG}}\end{matrix}$

Note that e=k₁ and ē=k_(S−1). Therefore, for a buy order for a collaredforward, the replicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{0\mspace{14mu} {for}\mspace{14mu} s} = 1} \\{{{{E\left\lbrack {U{k_{s - 1} \leq U < k_{s}}} \right\rbrack} - {k_{1}\mspace{14mu} {for}\mspace{14mu} s}} = 2},3,\ldots \mspace{14mu},{S - 1}} \\{{k_{S - 1} - {k_{1}\mspace{14mu} {for}\mspace{14mu} s}} = S}\end{matrix} \right.} & {10.2{.2}{AH}}\end{matrix}$

Note that the formulas for the a's are identical to those for a callspread with strikes of k₁ and k_(S−1).

For a sell order for a collared forwarded

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{k_{S - 1} - {k_{1}\mspace{14mu} {for}\mspace{14mu} s}} = 1} \\{{{k_{S - 1} - {{E\left\lbrack {U{k_{s - 1} \leq U < k_{s}}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s}} = 2},3,\ldots \mspace{14mu},{S - 1}} \\{{0\mspace{14mu} {for}\mspace{14mu} s} = S}\end{matrix} \right.} & {10.2{.2}{AI}}\end{matrix}$

Note that buys and sells of collared forwards, by construction, havebounded replication P&L.Replicating Digital Options with a Maximum Fixed Price

An auction sponsor can offer customers derivatives strategies where thecustomer specifies the maximum price to pay and then the strike isdetermined such that the customer pays as close as possible to but nogreater than the specified price. Offering these derivatives strategieswill allow a customer to trade such market strategies as the over-understrategy, where the customer receives a notional quantity equal to twicethe price. As another example, a customer could trade a digital optionwith a specific payout of say 5 to 1. These derivatives strategies mayprovide the customer with an option with a strike that may not beavailable for other options in the auction. For example, in general theoption strikes on these strategies may be different from k₁, k₂, . . . ,k_(S−1) thus these derivative strategies provide the customers withcustomized strikes.

For illustrative purposes, assume that the customer desires a digitalcall option and specifies a price p_(*), which is the maximum price thecustomer is willing to pay for this option. Based on this p_(*), thestrike k_(*) is then determined such that k_(*) is as low as possiblesuch that the price of the digital call option is no greater than p_(*).To implement an over-under strategy, the customer will submit a price ofp_(*)=0.5 and to request a digital option with a 5 to 1 payout, thecustomer will submit a price of p_(*)=0.2.

This digital call option pays out 1 if the underlying U is greater thanor equal to k_(*) and zero otherwise. Therefore,

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {{{for}\mspace{14mu} U} < k_{*}} \\1 & {{{for}\mspace{14mu} k_{*}} \leq U}\end{matrix} \right.} & {10.2{.2}{AJ}}\end{matrix}$

Assume that the digital call option struck at k_(v−1) has a price lessthan p_(*) and the digital call option struck at k_(v) has a pricegreater than or equal to p_(*) and assume that 1≦v≦S−2. Therefore,

k _(v−1) <k _(*) ≦k _(v)  10.2.2AK

In the special case where the price of the digital call option struck atk_(v) equals exactly p_(*), then k_(*)=k_(v) and the replicatingdigitals for this derivatives strategy are the replicating digitals forthe digital call struck at k_(v).

The expected value of the digital call option payout is

E[d(U)]=Pr[k _(*) ≦U]  10.2.2AL

The auction sponsor will set k_(*) such that the expected payout on theoption is as close to as possible but no greater than p_(*). Namely,k_(*) is the minimum value such that

Pr[k _(*) ≦U]≦p _(*)  10.2.2AM

Therefore, in terms of the states s, the payout of the option strategyis

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},{v - 1}} \\0 & {{{for}\mspace{14mu} s} = {{v\mspace{14mu} {and}\mspace{14mu} U} < k_{*}}} \\1 & {{{for}\mspace{14mu} s} = {{v\mspace{14mu} {and}\mspace{14mu} k_{*}} \leq U}} \\1 & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}{AN}}\end{matrix}$

Now, as for the previously considered digital options e=0 and ē=1. Bythe general replication theorem of appendix 10B then

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{0\mspace{14mu} {for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},{v - 1}} \\{{\frac{\Pr \left\lbrack {k_{*} \leq U < k_{v}} \right\rbrack}{p_{v}}\mspace{14mu} {for}\mspace{14mu} s} = v} \\{{{1\mspace{14mu} {for}{\mspace{11mu} \;}s} = {v + 1}},{v + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}{AO}}\end{matrix}$

The price of this digital call option can be computed as

$\begin{matrix}\begin{matrix}{{\sum\limits_{s = 1}^{S}{a_{s}p_{s}}} = {{\frac{\Pr \left\lbrack {k_{*} \leq U < k_{v}} \right\rbrack}{p_{v}}p_{v}} + {\sum\limits_{s = {v + 1}}^{S}p_{s}}}} \\{= {{\Pr \left\lbrack {k_{*} \leq U < k_{v}} \right\rbrack} + {\sum\limits_{s = {v + 1}}^{S}p_{s}}}} \\{= {\Pr \left\lbrack {k_{*} \leq U} \right\rbrack}} \\{\leq p_{*}}\end{matrix} & {10.2{.2}{AP}}\end{matrix}$

where the last step follows by how k_(*) is constructed.

By the general replication theorem of appendix 10B, the sell of thisstrategy has the following replicating digitals

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{1\mspace{14mu} {for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},{v - 1}} \\{{1 - {\frac{\Pr \left\lbrack {k_{*} \leq U < k_{v}} \right\rbrack}{p_{v}}\mspace{14mu} {for}\mspace{14mu} s}} = v} \\{{{0\mspace{14mu} {for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}{AQ}}\end{matrix}$

It is worth noting that the infimum replication P&L for buys and sellsof this strategy is finite.Replicating Vanilla Options with a Fixed Price

This section shows how to replicate a vanilla option with a fixed price.For illustrative purposes, assume that a customer requests to purchase avanilla call with a price of p_(*). Let k_(*) denote the strike price ofthe option to be determined to create an option with a price of p_(*).This vanilla call pays out as follows

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{{0\mspace{14mu} {for}\mspace{14mu} U} < k_{*}} \\{{U - {k_{*}\mspace{14mu} {for}\mspace{14mu} k_{*}}} \leq U}\end{matrix} \right.} & {10.2{.2}{AR}}\end{matrix}$

Assume that the price of a vanilla call with a strike of k_(v−1) is lessthan p_(*) and that the price of a vanilla call with a strike of k_(v)is greater than or equal to p_(*). Thus,

k _(v−1) <k _(*) ≦k _(v)  10.2.2AS

In the special case that the price of a vanilla call with a strike ofk_(v) is exactly p_(*) then set k_(*)=k_(v) and the replicating digitalsare the replicating digitals for a vanilla call.

Now, the expected value of the payout on this option is

$\begin{matrix}{{E\left\lbrack {d(U)} \right\rbrack} = {{{E\left\lbrack {{U - k_{*}}{k_{*} \leq U < k_{v}}} \right\rbrack}{\Pr \left\lbrack {k_{*} \leq U < k_{v}} \right\rbrack}} + {\sum\limits_{s = {v + 1}}^{S}{{E\left\lbrack {{U - k_{*}}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}p_{s}}}}} & {10.2{.2}{AT}}\end{matrix}$

Set k_(*) such that the expected payout on the option equals the pricep_(*). Namely, that

$\begin{matrix}{p_{*} = {{{E\left\lbrack {{U - k_{*}}{k_{*} \leq U < k_{v}}} \right\rbrack}{\Pr \left\lbrack {k_{*} \leq U < k_{v}} \right\rbrack}} + {\sum\limits_{s = {v + 1}}^{S}{{E\left\lbrack {{U - k_{*}}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}p_{s}}}}} & {10.2{.2}{AU}}\end{matrix}$

Solving for k_(*) in this equation may require a one-dimensionaliterative search.

Therefore, the derivatives strategy d in terms of states is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{{{0\mspace{14mu} {for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},{v - 1}} \\{{0\mspace{14mu} {for}\mspace{14mu} s} = {{v\mspace{14mu} {and}\mspace{14mu} U} < k_{*}}} \\{{U - {k_{*}\mspace{14mu} {for}\mspace{14mu} s}} = {{v\mspace{14mu} {and}\mspace{14mu} k_{*}} \leq U}} \\{{{U - {k_{*}\mspace{14mu} {for}\mspace{14mu} s}} = {v + 1}},{v + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}{AV}}\end{matrix}$

Note that e=0 and ē=E[U−k_(*)|k_(S−1)≦U]. Therefore, by the generalreplication theorem of appendix 10B, the replicating digitals are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{0\mspace{14mu} {for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},{v - 1}} \\{{\frac{{\Pr \left\lbrack {k_{*} \leq U < k_{v}} \right\rbrack}\left( {E\left\lbrack {{U - k_{*}}{k_{*} \leq U < k_{v}}} \right\rbrack} \right)}{p_{v}}\mspace{14mu} {for}\mspace{14mu} s} = v} \\{{{{E\left\lbrack {{U - k_{*}}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}{AW}}\end{matrix}$

To check that the replication of this derivatives strategy has a priceof p_(*), note that the price is

$\begin{matrix}\begin{matrix}{{\sum\limits_{s = 1}^{S}{a_{s}p_{s}}} = {{\frac{{\Pr \left\lbrack {k_{*} \leq U < k_{v}} \right\rbrack}\left( {E\left\lbrack {{U - k_{*}}{k_{*} \leq U < k_{v}}} \right\rbrack} \right)}{p_{v}}p_{v}} +}} \\{{\sum\limits_{s = {v + 1}}^{S}{p_{s}{E\left\lbrack {{U - k_{*}}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}}}} \\{= {{{\Pr \left\lbrack {k_{*} \leq U < k_{v}} \right\rbrack}{E\left\lbrack {{U - k_{*}}{k_{*} \leq U < k_{v}}} \right\rbrack}} +}} \\{{\sum\limits_{s = {v + 1}}^{S}{p_{s}{E\left\lbrack {{U - k_{*}}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}}}} \\{= p_{*}}\end{matrix} & {10.2{.2}{AX}}\end{matrix}$

For sells of this strategy,

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{{{E\left\lbrack {{U - k_{*}}{k_{s - 1} \leq U}} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},{v - 1}} \\{{{E\left\lbrack {{U - k_{*}}{k_{s - 1} \leq U}} \right\rbrack} - {\frac{\begin{matrix}{\Pr \left\lbrack {k_{*} \leq U < k_{v}} \right\rbrack} \\\left( {E\left\lbrack {{U - k_{*}}{k_{*} \leq U < k_{v}}} \right\rbrack} \right)\end{matrix}}{p_{v}}\mspace{14mu} {for}\mspace{14mu} s}} = v} \\{{{\begin{matrix}{{{E\left\lbrack {{U - k_{*}}{k_{s - 1} \leq U}} \right\rbrack} -}\;} \\{E\left\lbrack {{U - k_{*}}{k_{s - 1} \leq U < k_{s}}} \right\rbrack}\end{matrix}\mspace{11mu} {for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.2{.2}{AY}}\end{matrix}$

Using this approach, an auction sponsor can use digital options toreplicate a vanilla call option with a fixed delta. In this way, acustomer can request a vanilla call option with a 25 delta or a 50 deltasince the price of a vanilla call option is a one-to-one function of thedelta of a vanilla call option (with the option maturity, the forward ofthe underlying, the implied volatility as a differentiable function ofstrike, and the interest rate all fixed and known).

In addition to replicating vanilla call options, auction sponsors canuse this replication approach to offer fixed price options for, but notlimited to, vanilla puts, call spreads, and put spreads.

Summary of Replication P&L

Table 10.2.2-1 shows the replication P&L for these different derivativesstrategies discussed above.

TABLE 10.2.2-1 Replication P&L for different derivative strategies.Derivative Strategy Replication P&L A digital call 0 A digital put 0 Arange binary 0 A vanilla call Possibly Infinite A vanilla put PossiblyInfinite A call spread Finite A put spread Finite A straddle PossiblyInfinite A collared straddle Finite A forward Possibly Infinite Acollared forward Finite A digital call with maximum price Finite Avanilla call with a fixed price Possibly Infinite

10.2.3 Replicating Derivatives Strategies when the Underlying isTwo-Dimensional

Assume that the underlying U is two-dimensional and let U₁ and U₂ denoteone-dimensional random variables as follows

U=(U ₁ ,U ₂)  10.2.3A

Assume that derivatives strategies will be based on a total of S₁−1strikes for U₁ denoted k₁ ¹, k₂ ¹, k₃ ¹, k_(s) ₁ ⁻¹ ¹, and assume optionstrategies will be based on a total of S₂−1 strikes for U₂ denoted k₁ ²,k₂ ², k₃ ², k_(s) ₂ ⁻¹ ². Note that a superscript of 1 is used to denotestrikes associated with U₁ and a superscript of 2 is used to denotestrikes associated with U₂. Further, assume that

k ₁ ¹ <k ₂ ¹ <k ₃ ¹ < . . . <k _(S) ₂ ⁻² ¹ <k _(S) ₁ ⁻² ¹  10.2.3B

k ₁ ² <k ₂ ² <k ₃ ² < . . . <k _(S) ₂ ⁻² ² <k _(S) ₂ ⁻² ²  10.2.3C

Thus, the strikes are in ascending order based on the subscript.Further, for notational convenience, for U₁ let k₀ ¹=−∞ and let k_(S) ₁¹=∞. For U₂, let k₀ ²=−∞ and k_(s) ₂ ²=∞. These four variables do notrepresent actual strikes but will be useful in representing formulaslater.

For terminology, let state (i,j) denote an outcome U such that

[U:k _(i−1) ¹ ≦U ₁ <k _(i) ¹ ]∩[U:k _(j−1) ² ≦U ₂ <k _(j) ²]  10.2.3D

for i=1, 2, . . . , and j=1, 2, . . . , S₂.Let p_(ij) denote the probability of state (i,j) occurring. That is

p _(ij) =Pr[k _(i−1) ¹ ≦U ₁ <k _(i) ¹&k _(j−1) ² ≦U ₂ <k _(j)²]  10.2.3E

for i=1, 2, . . . , S₁ and j=1, 2, . . . , S₂. Let a_(ij) denote thereplicating quantity of digitals for state (i, j).

The remainder of the section will use this notation tailored to thistwo-dimensional case. However, it is worth mapping this notation intothe general notation from section 10.2.1. In this case one can enumeratea mapping from state (i,j) into state s as follows

s=(i−1)S ₂ +j for i=1, 2, . . . , S_(i) and j=1, 2, . . . , S₂  10.2.3F

This defines s for s=1, 2, . . . , S where S=S₁×S₂. Then

Ω_(s) =[U:k _(i−1) ¹ ≦U ₁ <k _(i) ¹ ,k _(j−1) ² ≦U ₂ <k _(j) ²]  10.2.3G

p_(s)=p_(ij)  10.2.3H

a_(s)=a_(ij)  10.2.3I

for s=(i−1)S₂+j.

The general replication theorem from appendix 10B can be used to deriveresults for this two-dimensional case. The digital replication for a buyis

a _(ij) =E[d(U ₁ ,U ₂)|k _(i−1) ¹ ≦U ₁ <k _(i) ¹&k _(j−1) ² ≦U ₂ <k _(j)² ]−e   10.2.3J

-   -   for i=1, 2, . . . , S₁ and j=1, 2, . . . , S₂        where

$\begin{matrix}{\underset{\_}{e} = {\underset{{j = 1},2,\ldots \mspace{14mu},S_{2}}{\min\limits_{{i = 1},2,\ldots \mspace{14mu},S_{1}}}{E\left\lbrack {d\left( {U_{1},U_{2}} \right)} \middle| {k_{i - 1}^{1} \leq U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}\mspace{14mu} k_{j - 1}^{2}} \leq U_{2} < k_{j}^{2}} \right\rbrack}}} & {10.2{.3}K}\end{matrix}$

The replication P&L and the infimum replication P&L for a buy is givenby

$\begin{matrix}{C = {\sum\limits_{i = 1}^{S_{1}}\; {\sum\limits_{j = 1}^{S_{2}}{{I\left\lbrack {k_{i - 1}^{1} \leq U_{1} < k_{i}^{1}} \right\rbrack}{I\left\lbrack {k_{j - 1}^{2} \leq U_{2} < k_{j}^{2}} \right\rbrack} \times \times \left( {{E\left\lbrack {d\left( {U_{1},U_{2}} \right)} \middle| {k_{i - 1}^{1} \leq U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}\mspace{14mu} k_{j - 1}^{2}} \leq U_{2} < k_{j}^{2}} \right\rbrack} - {d\left( {U_{1},U_{2}} \right)}} \right)}}}} & {10.2{.3}L} \\{{\inf \mspace{14mu} C} = {\underset{{j = 1},2,\ldots \mspace{14mu},S_{2}}{\min\limits_{{i = 1},2,\ldots \mspace{14mu},S_{1}}}\left\lbrack {\inf\limits_{\substack{k_{i - 1}^{1} \leq U_{1} < k_{i}^{1} \\ k_{j - 1}^{2} \leq U_{2} < k_{j}^{2}}}\left( \begin{matrix}{{E\begin{bmatrix}\left. {d\left( {U_{1},U_{2}} \right)} \middle| {k_{i - 1}^{1} \leq} \right. \\{U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}\mspace{14mu} k_{j - 1}^{2}} \leq U_{2} < k_{j}^{2}}\end{bmatrix}} -} \\{d\left( {U_{1},U_{2}} \right)}\end{matrix} \right)} \right\rbrack}} & {10.2{.3}M}\end{matrix}$

For sells of the strategy based on d

a _(ij) =ē−E[d(U)|k _(i−1) ¹ ≦U ₁ <k _(i) ¹&k _(j−1) ² ≦U ₂ <k _(j) ²]  10.2.3N

-   -   for i=1, 2, . . . , S₁ and j=1, 2, . . . , S₂        where

$\begin{matrix}{\overset{\_}{e} = {\underset{{j = 1},2,\ldots \mspace{11mu},S_{2}}{\min\limits_{{i = 1},2,\ldots \mspace{11mu},S_{1}}}{E\left\lbrack {d\left( {U_{1},U_{2}} \right)} \middle| {k_{i - 1}^{1} \leq U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}\mspace{14mu} k_{j - 1}^{2}} \leq U_{2} < k_{j}^{2}} \right\rbrack}}} & {10.2{.3}O}\end{matrix}$

The replication P&L, and the infimum replication P&L are

$\begin{matrix}{C = {\sum\limits_{i = 1}^{S_{1}}\; {\sum\limits_{j = 1}^{S_{2}}\; {{I\left\lbrack {k_{i - 1}^{1} \leq U_{1} < k_{i}^{1}} \right\rbrack}{I\left\lbrack {k_{j - 1}^{2} \leq U_{2} < k_{j}^{2}} \right\rbrack} \times \times \left( {{d\left( {U_{1},U_{2}} \right)} - {E\left\lbrack {d\left( {U_{1},U_{2}} \right)} \middle| {k_{i - 1}^{1} \leq U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}\mspace{14mu} k_{j - 1}^{2}} \leq U_{2} < k_{j}^{2}} \right\rbrack}} \right\rbrack \left. \quad \right)}}}} & {10.2{.3}P} \\{{\inf \mspace{11mu} C} = {\underset{{j = 1},2,\ldots \mspace{14mu},S_{2}}{\min\limits_{{i = 1},2,\ldots \mspace{14mu},S_{1}}}\left\lbrack {\inf\limits_{\substack{k_{i - 1}^{1} \leq U_{1} < k_{i}^{1} \\ k_{j - 1}^{2} \leq U_{2} < k_{j}^{2}}}\begin{pmatrix}{{d\left( {U_{1},U_{2}} \right)} -} \\{E\begin{bmatrix}\left. {d\left( {U_{1},U_{2}} \right)} \middle| {k_{i - 1}^{1} \leq U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}}} \right. \\{k_{j - 1}^{2} \leq U_{2} < k_{j}^{2}}\end{bmatrix}}\end{pmatrix}} \right\rbrack}} & {10.2{.3}Q}\end{matrix}$

The variance of the replication P&L for both buys and sells of d is

$\begin{matrix}{{{Var}\lbrack C\rbrack} = {\sum\limits_{i = 1}^{S_{1}}{\sum\limits_{j = 1}^{S_{2}}{p_{ij}{{Var}\left\lbrack {d\left( {U_{1},U_{2}} \right)} \middle| {k_{i - 1}^{1} \leq U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}\mspace{14mu} k_{j - 1}^{2}} \leq U_{2} < k_{j}^{2}} \right\rbrack}}}}} & {10.2{.3}R}\end{matrix}$

Replicating Derivatives Strategies that Depend Upon Only One Underlying

With a two-dimensional underlying (or equivalently option strategiesbased on two univariate random variables), an auction sponsor can offercustomers option strategies described above from the one-dimensionalunderlying including but not limited to call spreads and put spreads.Including these univariate vanilla options with a two-dimensionalunderlying will help aggregate liquidity in these markets as it allowscustomers to take positions based on U₁ individually, U₂ individually,or U₁ and U₂ jointly all in the same auction.

To see how this can be done, as an illustration consider a call spreadwith strikes k_(v) ¹ and k_(w) ¹. To price a call spread on U₁ in thisframework define

$\begin{matrix}{{d\left( {U_{1},U_{2}} \right)} = \left\{ \begin{matrix}{{0\mspace{14mu} {if}\mspace{14mu} U_{1}} < k_{v}^{1}} \\{{U_{1} - {k_{v}^{1}\mspace{14mu} {if}\mspace{14mu} k_{v}^{1}}} \leq U_{1} < k_{w}^{1}} \\{{k_{w}^{1} - {k_{v}^{1}\mspace{14mu} {if}\mspace{14mu} k_{w}^{1}}} \leq U_{1}}\end{matrix} \right.} & {10.2{.3}S}\end{matrix}$

Note that this function does not depend upon U₂ in any way. For a buy ofd in this case, the replicating digitals are

$\begin{matrix}{a_{ij} = \left\{ \begin{matrix}{{{0\mspace{14mu} {for}\mspace{14mu} i} = 1},2,\ldots \mspace{14mu},{{v\mspace{14mu} {and}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},S_{2}} \\{{E\left\lbrack U_{1} \middle| {k_{i - 1}^{1} \leq U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}\mspace{14mu} k_{j - 1}^{2}} \leq U_{2} < k_{j}^{2}} \right\rbrack} - k_{v}^{1}} \\{{{{for}\mspace{14mu} i} = {v + 1}},{v + 2},\ldots \mspace{14mu},{{w\mspace{14mu} {and}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},S_{2}} \\{{{k_{w}^{1} - {k_{v}^{1}\mspace{14mu} {for}\mspace{14mu} i}} = {w + 1}},{w + 2},\ldots \mspace{14mu},{{S_{1}\mspace{14mu} {and}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},S_{2}}\end{matrix} \right.} & {10.2{.3}T}\end{matrix}$

Also, for a sell order on d

$\begin{matrix}{a_{ij} = \left\{ \begin{matrix}{{{k_{w}^{1} - {k_{v}^{1}\mspace{14mu} {for}\mspace{14mu} i}} = 1},2,\ldots \mspace{14mu},{{v\mspace{14mu} {and}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},S_{2}} \\{k_{w}^{1} - {E\left\lbrack U_{1} \middle| {k_{i - 1}^{1} \leq U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}\mspace{14mu} k_{j - 1}^{2}} \leq U_{2} < k_{j}^{2}} \right\rbrack}} \\{{{{for}\mspace{14mu} i} = {v + 1}},{v + 2},\ldots \mspace{14mu},{{w\mspace{14mu} {and}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},S_{2}} \\{{{0\mspace{14mu} {for}\mspace{14mu} i} = {w + 1}},{w + 2},\ldots \mspace{14mu},{{S_{1}\mspace{14mu} {and}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},S_{2}}\end{matrix} \right.} & {10.2{.3}U}\end{matrix}$

For a specific i, consider the case where

E[U ₁ |k _(i−1) ¹ ≦U ₁ <k _(i) ¹&k _(j−1) ² ≦U ₂ <k _(j) ² ]=E[U ₁ |k_(i−1) ¹ ≦U ₁ <k _(i) ¹]  10.2.3V

for all j, i.e. if the conditional expectation of U₁ given i and j isequal to the conditional expectation of U₁ given i. This condition willbe satisfied, for instance, if U₁ and U₂ are independent randomvariables. Then under this condition the formula for a_(ij) for buys andsells simplifies to the replication formulas for a call spread inone-dimension discussed in section 10.3. Specifically, equation 10.2.3Tsimplifies to equation 10.2.2Q, and equation 10.2.3U simplifies toequation 10.2.2R.

Replicating Derivatives Strategies on the Sum, Difference, Product andQuotient of Two Variables

To create an option on the sum of two variables, set the function d asfollows

$\begin{matrix}{{d\left( {U_{1},U_{2}} \right)} = \left\{ \begin{matrix}{{{0\mspace{14mu} {if}\mspace{14mu} U_{1}} + U_{2}} < k} \\{{U_{1} + U_{2} - {k\mspace{14mu} {if}\mspace{14mu} k}} \leq {U_{1} + U_{2}}}\end{matrix} \right.} & {10.2{.3}W}\end{matrix}$

Assume that the strikes are all non-negative and the underlyings arealso non-negative (so k₀ ¹=0 and k₀ ²=0) and further assume that k>k₁¹+k₁ ². In this case, for a buy of d

a _(ij) =E[max(U ₁ +U ₂ k,0)|k _(i−1) ¹ ≦U ₁ <k _(i) ¹&k _(j−1) ² ≦U ₂<k _(j) ²]  10.2.3X

-   -   for i=1, 2, . . . , S₁ and j=1, 2, . . . , S₂        Note that several values of a_(ij) will likely be zero in this        case. For example, since k>k₁ ¹+k₁ ², then in state (1, 1)        U₁+U₂<k so a₁₁=0. This implies that e=0. For a sell of d, then,

a _(ij) =ē−E[max(U ₁ +U ₂ −k,0)|k _(i−1) ¹ ≦U ₁ <k _(i) ¹& k _(j−1) ² ≦U₂ <k _(j) ²]  10.2.3Y

-   -   for i=1, 2, . . . , S₁ and j=1, 2, . . . , S₂        where

ē=E[U ₁ +U ₂ k|k _(s) ₁ ⁻¹ ¹ ≦U ₁&k _(s) ₂ ⁻¹ ¹ ≦U ₂]  10.2.3Z

Creating options on the sum of two variables will be useful to theauction sponsor. For example, U₁ could be the number of heating degreedays for January and U₂ could be the number of heating degree days forFebruary. Then the sum of these two variables U₁+U₂ is the number ofheating degree days for January and February combined.

In a similar fashion an auction sponsor can offer derivatives strategieson the difference between U₁ and U₂. For instance, U₁ could be the levelof target federal funds at the end of the next federal reserve openmarket committee meeting and U₂ could be level of target federal fundsafter the following such meeting. Thus an option on the difference U₂−U₁would relate to what happens between the end of the 1^(st) meeting andthe end of the 2^(nd) meeting. In addition, derivatives strategies ondifferences can be applied to the interest rate market. If U₁ is atwo-year interest rate at the close at a certain date in the future andU₂ is a ten-year interest rate at the close at the same future date,then the difference represents the slope of the interest rate curve atthe future specified date. If U, is the yield on a 10-year referenceTreasury at the close at a certain date in the future and U₂ is the10-year swap rate at the close at the same date in the future, then thedifference represents the swaps spread.

In a similar fashion, an auction sponsor can create options on theproduct of two variables. For example if U₁ is the exchange rate ofdollars per euro and U₂ is the exchange rate of yen per dollar; thenU₁×U₂ is the exchange rate of yen per euros.

Further, an auction sponsor can create options on the quotient of twovariables. In the foreign exchange market, if U₁ is the Canadian dollarexchange rate per US dollar and U₂ is the Japanese yen exchange rate perdollar then U₂/U₁ is the cross rate or the Japanese yen per Canadiandollar exchange rate. As another example, if U₂ is the price of a stock,U₁ is the earnings on a stock. Then U₂/U₁ is the price earnings multipleof the stock.

Note that U₁ and U₂ in the examples described above have both been basedon similar variables such as both based on weather outcomes. However,there is no requirement that U₁ and U₂ be closely related: in fact, theycan represent underlyings that bear little or no relation to oneanother. For example U₁ may represent an underlying based on weather andU₂ may be an underlying based on a foreign exchange rate.

Replicating a Path Dependent Option

An example embodiment in two-dimensions can offer customers the abilityto trade path dependent options. For example, consider a call optionwith an out-of-the-money knock out. Namely, the option pays out if theminimum of the exchange rate remains above a certain barrier k_(v) andspot is above the strike k_(w) on expiration.

Let X_(t) denote the exchange rate of a currency per dollar at time t.Let U₁ denote the minimum value of the exchange rate over a time periodso that

U₁=Min{X_(t),0≦t≦T}  10.2.3AA

where T denotes the expiration of the option. Let U₂ denote X_(T), theexchange rate at time T. Then the derivatives strategy pays out asfollows

$\begin{matrix}{{d\left( {U_{1},U_{2}} \right)} = \left\{ \begin{matrix}{{0\mspace{14mu} {if}\mspace{14mu} U_{1}} < {k_{v}\mspace{14mu} {or}\mspace{14mu} U_{2}} < k_{w}} \\{{U_{2} - {k_{w}\mspace{14mu} {if}\mspace{14mu} k_{v}}} \leq {U_{1}\mspace{14mu} {and}\mspace{14mu} k_{w}} \leq U_{2}}\end{matrix} \right.} & {10.2{.3}{AB}}\end{matrix}$

Therefore, the replicating digitals are

$\begin{matrix}{a_{ij} = \left\{ \begin{matrix}{{{0\mspace{14mu} {for}\mspace{14mu} i} = 1},2,\ldots \mspace{14mu},{{v\mspace{14mu} {and}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},S_{2}} \\{{E\left\lbrack U_{2} \middle| {k_{i - 1}^{1} \leq U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}\mspace{14mu} k_{j - 1}^{2}} \leq U_{2} < k_{j}^{2}} \right\rbrack} - k_{w}} \\{{{{for}\mspace{14mu} i} = {v + 1}},{v + 2},\ldots \mspace{14mu},{{S_{1}\mspace{14mu} {and}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},S_{2}}\end{matrix} \right.} & {10.2{.3}{AC}}\end{matrix}$

For a sell of this option strategy,

$\begin{matrix}{a_{ij} = \left\{ \begin{matrix}{\overset{\_}{e} - {E\left\lbrack U_{2} \middle| {k_{i - 1}^{1} \leq U_{1} < {k_{i}^{1}\mspace{14mu} \text{\&}\mspace{14mu} k_{j - 1}^{2}} \leq U_{2} < k_{j}^{2}} \right\rbrack}} \\{{{{for}\mspace{14mu} i} = 1},2,\ldots \mspace{14mu},{{v\mspace{14mu} {and}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},S_{2}} \\{{{0\mspace{14mu} {for}\mspace{14mu} i} = {v + 1}},{v + 2},\ldots \mspace{14mu},{{S_{1}\mspace{14mu} {and}\mspace{14mu} j} = 1},2,\ldots \mspace{14mu},S_{2}}\end{matrix} \right.} & {10.2{.3}{AD}}\end{matrix}$

It is worth noting that the replicating digitals depend on the quantity

E[U ₂ |k _(i−1) ¹ ≦U ₁ <k _(i) ¹&k _(j−1) ² ≦U ₂ <k _(j) ²]  10.2.3AE

which is equal to

E[X _(T) |k _(i−1) ¹≦Min{X _(t),0≦t≦T}<k _(i) ¹ ,k _(j−1) ¹ <X _(T) <k_(j) ¹]  10.2.3AF

Note that Min{X_(t), 0≦t≦T} and X_(T) are in general not independentquantities: for example, they will be positively correlated if the pathof the exchange rate follows a Brownian motion. Thus this conditionalexpectation may require methods that incorporate this correlation tocompute this quantity.

Note that this strategy d has unbounded replication P&L. The auctionsponsor could offer customers a knock out call spread to allow forstrategies with bounded replication P&L.

10.2.4 Replicating Derivatives Strategies Based on Three or MoreVariables

Using the general formulas in section 10.1 such as equation 10.1I, anauction sponsor could replicate other types of derivatives strategies.To trade out of the money knockout options and in the money knockoutoptions, one could set U₁ to be the minimum over a time period, U₂ to bea maximum over a time period, and U₃ to represent the closing value overthe time period.

Section 10.3 Estimating the Distribution of the Underlying U

Sections 10.1 and 10.2 describe how to replicate derivatives strategiesusing digital options. This replication technique depends on certainaspects of the distribution of the underlying U. For example, thereplicating digitals and the replication variance for vanilla optionsdepend upon E[U|k_(s−1)≦U<k_(s)] and Var[U|k_(s−1)≦U<k_(s)]. Thissection shows how different example embodiments can be used to computethese conditional moments and, more generally, the distribution of U.The formulations in this Section 10.3, can be used and apply equally todetermine the conditional moments and the distributions of U forembodiments in which the replicating basis is the vanilla replicatingbasis (using replicating vanillas alone or together with replicatingdigitals) discussed in Section 11.2 et seq., as opposed to being thedigital replicating basis (using replicating digitals alone) discussedin this Section 10 and in Section 11.1.

Techniques for estimating the distribution of U can be broadly dividedinto global approaches and local approaches. In a global approach, asingle parametric distribution is fitted or hypothesized for thedistribution of the underlying. An example of a global approach would beto use a normal distribution or log-normal distribution to model theunderlying. In contrast, in a local approach several distributions maybe combined together to fit the underlying. For instance, a localapproach may use a different distribution for each state in the samplespace.

Independent of whether the auction sponsor uses a global or localapproach, the auction sponsor has to choose whether or not to estimatethe replicating digitals based on the auction prices. Allowing thereplicating digitals to depend on the auction prices may help keep theconditional mean of the replication P&L equal to zero (where the mean isconditioned on the auction prices), as these replicating digitals willbe based on the market determined distribution for U. However, thisdependence on auction prices adds iterations to the calculation engineas follows: when the equilibrium prices change due to say a new order,then the replication amounts for each order will change, which will thenchange the equilibrium prices, which will again change the replicationamounts, and so on. This process will slow down the calculation ofequilibrium prices. On the other hand, if the replicating digitals areconstant through the auction and do not depend on the auction prices,then convergence techniques will not require this extra iteration butthe replication P&L may not have a conditional mean equal to zero, giventhe auction prices.

If the auction sponsor wants to offer customers fixed priced options asdiscussed in section 10.2.2, then the auction sponsor will have to adoptthe more computationally intensive technique to compute the equilibrium,since the set of replicating digitals depend upon the auction prices forthese options.

This section begins with a discussion of the global approach in section10.3.1 followed by a discussion of the local approach in 10.3.2.

10.3.1 The Global Approach

This section discusses how the auction sponsor can use the globalapproach for estimating the distribution of U. First, this sectiondescribes how the auction sponsor selects a distribution for theunderlying. Second, this section describes how the auction sponsorestimates the parameters of that distribution. Then, this section showshow the auction sponsor can compute the replicating digitals after theparameters of the distribution are estimated. This section concludeswith an illustrative example.

Classes of Distributions for the Underlying

The auction sponsor may assume that the underlying follows a log-normaldistribution, a distribution that is used frequently when the underlyingis the price of a financial asset or for other variables that can onlytake on positive values. The log-normal model is used, for example, inthe Black-Scholes pricing formula. The auction sponsor may model theunderlying to be normally distributed, a distribution that has beenshown to approximate many variables. In addition if the underlying isthe continuously compounded return on an asset, then the return will benormally distributed if the price of the asset is log-normallydistributed.

The auction sponsor may choose a distribution that matches specificcharacteristics of the distribution of the underlying. If the underlyinghas fatter tails than the normal distribution, then the auction sponsormay model the underlying as t distributed. If the underlying haspositive skewness, then the auction sponsor might model the underlyingas gamma distributed. If the underlying has time-varying volatility,then the auction sponsor may model the underlying as a GARCH process.

In addition to the continuous distributions described above, the auctionsponsor may model the underlying using a discrete distribution, sincemany underlyings may in fact take on only a discrete set of values. Forexample, US CPI is reported to the nearest tenth and heating degree daysare typically reported to the nearest degree, so both of these arediscrete random variables.

To handle discreteness, the auction sponsor may model U as a discreterandom variable such as a multinomial random variable. In other exampleembodiments, the auction sponsor may choose to discretize a continuousrandom variable. For notation, let ρ denote the level of precision towhich that the underlying U is reported. For example, ρ equals 0.1 ifthe underlying U is US CPI and ρ equals 1 if the underlying U is heatingdegree days. To model U as a discretized random variable let V denote acontinuous distributed random variable and let

$\begin{matrix}{U = {{R\left( {V,\rho} \right)} = {\rho \times {{int}\left\lbrack {\frac{V}{\rho} + 0.5} \right\rbrack}}}} & {10.3{.1}A}\end{matrix}$

where “int” represents the greatest integer function. U is discretizedthrough the function R applied to the continuous random variable V.

Selecting the Appropriate Distribution

The auction sponsor may select the distribution using a variety oftechniques. First, the choice of the distribution may be dictated byfinancial theory. For example, as in the Black-Scholes formula, thelog-normal distribution is often used when U denotes the price of afinancial instrument. Because of this, the normal distribution is oftenused when U denotes the return on the financial instrument.

If historical data on the underlying is available, the auction sponsorcan perform specification tests to determine a distribution that fitsthe historical data. For example, the auction sponsor may use historicaldata to compute excess kurtosis to test whether the normal distributionfits as well as the t distribution for U. As another example, theauction sponsor may use historical data to test for GARCH effects to seeif a GARCH model would best fit the data. If the underlying is adiscretized version of a continuous distribution, then the specificationtests may specifically incorporate this information.

Estimating the Parameters of the Distribution

The auction sponsor may estimate the parameters of the distributionusing a variety of approaches.

If historical data is available on the underlying, then the auctionsponsor can estimate the parameters of the distribution using techniquessuch as moment matching and maximum likelihood. If the variable isdiscrete, then this discreteness may be modeled explicitly using maximumlikelihood.

If options are traded on the underlying, then the auction sponsor canuse these option prices to estimate the distribution of the underlying.A large body of academic literature uses the prices on options toestimate the distribution of the underlying. In these methods theimplied volatility is expressed as a function of the option's strikeprice and then numerical derivatives are used to determine thedistribution of the underlying. For example, if the 25 delta calls havea higher implied volatility than the 75 delta calls, then this methodwill likely imply a negative skewness in the distribution of theunderlying.

If market economists or analysts forecast the underlying, the auctionsponsor can use these forecasts to help determine the mean and standarddeviation of the underlying. For example, when U represents an upcomingeconomic data release in the US such as nonfarm payrolls, between 20 and60 economists will often forecast the release. The mean and standarddeviation of these forecasts for instance may provide accurate estimatesof the mean and standard deviation of the underlying. As anotherexample, many equity analysts forecast the earnings for US largecompanies so if the underlying is the quarterly earnings of a largecompany, analyst forecasts can be used to estimate the parameters of thedistribution.

In addition, an auction sponsor may determine parameters of thedistribution based on the auction's implied distribution. In this case,an example embodiment may set the parameters of the distribution suchthat the implied probabilities of each state based on the distributionis close to or equal to the implied probabilities based on the auction'sdistribution.

Computing Replication Quantities from the Distribution

Once the auction sponsor has determined the distribution and theparameters of the distribution, the auction sponsor can then compute thequantities for the digital replication. For example, the quantity ofreplicating digitals for many option strategies such as vanilla optionsdepend on E[U|k_(s−1)≦U<k_(s)] and the replication variance for theseoptions depend upon Var[U|k_(s−1)≦U<k_(s)]. This section shows how toevaluate these quantities.

Consider the case where U is normally distributed with a mean μ andstandard deviation σ. Since U is a continuous random variable, therounding parameter ρ equals 0. Let the option strikes be denoted ask_(s) for s=1, 2, . . . , S−1. Appendix 10C shows that for s=2, 3, . . ., S−1

$\begin{matrix}{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = {\mu + \frac{\frac{\sigma}{\sqrt{2\pi}}\begin{bmatrix}{{\exp\left( {- \frac{\left( {k_{s - 1} - \mu} \right)^{2}}{2\sigma^{2}}} \right)} -} \\{\exp\left( {- \frac{\left( {k_{s} - \mu} \right)^{2}}{2\sigma^{2}}} \right)}\end{bmatrix}}{{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}}}} & {10.3{.1}B}\end{matrix}$

where “exp” denotes the exponential function or raising the argument tothe power of e. In this case, the variance of replication P&L willdepend upon Var[U|k_(s−1)≦U<k_(s)], which is equal to

$\begin{matrix}\begin{matrix}{{{Var}\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = {{E\left\lbrack U^{2} \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} -}} \\{\left( {E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} \right)^{2}} \\{= {\frac{\int_{k_{s - 1}}^{k_{s}}{u^{2}{f_{\mu,\sigma}(u)}{u}}}{\Pr \left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack} -}} \\{\left( {E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} \right)^{2}}\end{matrix} & {10.3{.1}C}\end{matrix}$

where ƒ_(μ,σ) denotes the normal density function with mean μ andstandard deviation σ. To evaluate this expression, the integral can becomputed using for example numerical techniques.

Next consider the case where U is a discretized normal. That is, let Vbe normally distributed with a mean μ and standard deviation a and let Ube a function of V as follows

U=R(V,ρ)  10.3.1D

where R is defined in equation 10.3.1A. In this case, all outcomes of Uare divisible by ρ. Assume that each strike k_(s) is exactly equal to apossible outcome of U and then for s=2, 3, . . . , S−1

$\begin{matrix}\begin{matrix}{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = {E\left\lbrack {R\left( {V,\rho} \right)} \middle| {k_{s - 1} \leq {R\left( {V,\rho} \right)} < k_{s}} \right\rbrack}} \\{= \frac{\sum\limits_{v = k_{s - 1}}^{k_{s} - \rho}{v\; {\Pr \left\lbrack {{v - {\rho/2}} \leq V < {v + {\rho/2}}} \right\rbrack}}}{\Pr \left\lbrack {{k_{s - 1} - {\rho/2}} \leq V < {k_{s} - {\rho/2}}} \right\rbrack}} \\{= \frac{\sum\limits_{v = k_{s - 1}}^{k_{s} - \rho}{v\; {\Pr \left\lbrack {{v - {\rho/2}} \leq V < {v + {\rho/2}}} \right\rbrack}}}{{N\left\lbrack \frac{k_{s} - \mu - \left( {\rho/2} \right)}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu - \left( {\rho/2} \right)}{\sigma} \right\rbrack}}} \\{= \frac{\sum\limits_{v = k_{s - 1}}^{k_{s} - \rho}{v\begin{pmatrix}{{N\left\lbrack \frac{v - \mu + \left( {\rho/2} \right)}{\sigma} \right\rbrack} -} \\{N\left\lbrack \frac{v - \mu - \left( {\rho/2} \right)}{\sigma} \right\rbrack}\end{pmatrix}}}{{N\left\lbrack \frac{k_{s} - \mu - \left( {\rho/2} \right)}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu - \left( {\rho/2} \right)}{\sigma} \right\rbrack}}}\end{matrix} & {10.3{.1}E}\end{matrix}$

where the summation variable v increases in increments of ρ.

Recall that the replication variance depends on Var[U|k_(s−1)≦U<k_(s)],which is equal to

$\begin{matrix}\begin{matrix}{{{Var}\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = {{E\left\lbrack U^{2} \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} -}} \\{\left( {E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} \right)^{2}} \\{= \frac{\sum\limits_{v = k_{s - 1}}^{k_{s} - \rho}{{v\;}^{2}{\Pr \left\lbrack {{v - {\rho/2}} \leq V < {v + {\rho/2}}} \right\rbrack}}}{\Pr \left\lbrack {{k_{s - 1} - {\rho/2}} \leq V < {k_{s} - {\rho/2}}} \right\rbrack}} \\{{- \left( {E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} \right)^{2}}} \\{= {\frac{\sum\limits_{v = k_{s - 1}}^{k_{s} - \rho}{v^{2}\begin{pmatrix}{{N\left\lbrack \frac{v - \mu + \left( {\rho/2} \right)}{\sigma} \right\rbrack} -} \\{N\left\lbrack \frac{v - \mu - \left( {\rho/2} \right)}{\sigma} \right\rbrack}\end{pmatrix}}}{\Pr \left\lbrack {{k_{s - 1} - {\rho/2}} \leq V < {k_{s} - {\rho/2}}} \right\rbrack} -}} \\{\left( {E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} \right)^{2}}\end{matrix} & {10.3{.1}F}\end{matrix}$

Example for Computing the Distribution and Replicating Digitals

Consider the following example to compute the replicating digitals foran auction using the global approach. Assume that the auction sponsorruns an auction for the change in US nonfarm payrolls for October 2001as released on Nov. 2, 2001. This example will show how economistforecasts can be used to create the replicating digitals. The underlyingU is measured in the change in the thousands of number of employed so anunderlying value of 100 means a payroll change of 100,000 people. Thepayrolls are rounded to the nearest thousand: since the underlying is inthousands, then ρ=1.

Table 10.3.1-1 shows forecasts from 55 economists surveyed by Bloombergfor this economic release. These forecasts have a mean of −299.05thousand people with a standard deviation of 70.04 thousand people.

TABLE 10.3.1-1 Economist forecasts for October 2001 change in US nonfarmpayrolls in thousands of people. −500 −400 −400 −400 −400 −385 −380 −380−360 −350 −350 −350 −350 −350 −350 −340 −325 −325 −325 −325 −300 −300−300 −300 −300 −300 −300 −300 −300 −300 −300 −300 −290 −289 −285 −283−275 −275 −275 −275 −275 −275 −275 −266 −250 −250 −250 −225 −210 −200−185 −150 −150 −150 −145If the auction sponsor assumes that U is a discretized version of thenormal, then the likelihood function is

$\begin{matrix}{{{Likelihood}\mspace{14mu} {Function}} = {\prod\limits_{t = 1}^{53}\left( {{N\left\lbrack \frac{f_{t} - \mu + \left( {\rho/2} \right)}{\sigma} \right\rbrack} - {N\left\lbrack \frac{f_{t} - \mu - \left( {\rho/2} \right)}{\sigma} \right\rbrack}} \right)}} & {10.3{.1}G}\end{matrix}$

where ƒ_(t) denotes the forecast from the t-th economist. The maximumlikelihood estimators give a mean of −299.06 and a standard deviation of69.40. Note that the maximum likelihood estimates are quite close to thesample mean and standard deviation, suggesting that the roundingparameter ρ is a small factor in the maximum likelihood estimation.

For this auction, the strikes are set to be −425, −375, −325, −275, −225and −175. Table 10.3.1-2 shows the values of Pr[k_(s−1)≦U<k_(s)],E[U|k_(s−1)≦U<k_(s)], and Var[U|k_(s−1)≦U<k_(s)] based on the model thatthe outcome is a discretized version of the normal with mean −299.06with a standard deviation of 69.40 and rounding to the nearest integer(ρ=1). This model is referred to as the discretized normal model.

TABLE 10.3.1-2 The probabilities, the conditional mean, and theconditional variance for the discretized normal model. State 2 State 3State 4 State 1 −425 <= U < −375 <= U < −325 <= U < State 5 State 6State 7 U < −425 −375 −325 −275 −275 <= U < −225 −225 <= U < −175 −175<= U Probability of State 0.0342 0.1011 0.2162 0.2813 0.2225 0.10710.0375 Conditional Expectation: −452.01 −396.26 −348.32 −300.44 −252.55−204.62 −147.75 E[U|state s] Conditional Variance: 609.13 194.14 201.88204.67 202.18 194.70 618.05 Var[U|state s]Based on this model one can compute the replicating digitals fordifferent derivatives strategies. Table 10.3.1-3 shows these replicatingdigitals, the prices of the strategies, and the variance of replicationP&L.

TABLE 10.3.1-3 The replicating digitals, prices, and variances ofdifferent strategies based on the global normal model. State 1 State 2State 3 State 4 State 5 State 6 State 7 U < −425 <= U < −375 <= U < −325<= U < −275 <= U < −225 <= U < −175 <= Price of Replication DerivativeStrategy −425 −375 −325 −275 −225 −175 U Strategy Variance Buy a digitalcall struck at −325 0.00 0.00 0.00 1.00 1.00 1.00 1.00 0.6484 0 Buy adigital put struck at −275 1.00 1.00 1.00 1.00 0.00 0.00 0.00 0.6329 0Buy a range binary with strikes 0.00 0.00 1.00 1.00 1.00 0.00 0.000.7200 0 of −375 and −225 Buy a vanilla call struck at −325 0.00 0.000.00 24.56 72.45 120.38 177.25 42.5715 146.60 Buy a vanilla put struckat −275 177.01 121.26 73.32 25.44 0.00 0.00 0.00 41.3320 141.70 Buy acall spread strikes at −375 0.00 0.00 26.68 74.56 122.45 150.00 150.0075.6798 146.22 and −225 Buy a put spread strikes at −375 150.00 150.00123.32 75.44 27.55 0.00 0.00 74.3202 146.22 and −225

10.3.2 Local Approach

In addition to the global approach described above, an auction sponsorcan apply a local approach where the underlying is modeled with a largenumber of parameters. In particular, the local approach can be set up tohave more parameters than states, whereas the global approach typicallyonly has one or two parameters. The local approach allows the auctionsponsor to fit the distribution of U with great flexibility.

The Intrastate Uniform Model

Assume that the distribution of U is discrete and that given that U isbetween k_(s−1) and k_(s), U is equally likely to be any of the possibleoutcomes within that state. In other words, if U is between k_(s−1) andk_(s), then U takes on the values

k_(s−1),k_(s−1)+ρ,k_(s−1)+2ρ, . . . , k_(s)−ρ  10.3.2A

and

Pr[U=k _(s−1) ]=Pr[U=k _(s−1) +ρ]= . . . =Pr[U=k _(s)−ρ]  10.3.2B

This intrastate uniform model can be used to compute the replicatingdigitals and the variance of the replicating digitals.

The conditional mean and the conditional variance for the intrastateuniform model are for s=2, 3, . . . , S−1

$\begin{matrix}{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = \frac{k_{s - 1} + k_{s} - \rho}{2}} & {10.3{.2}C} \\{{{Var}\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = \frac{\left( {k_{s} - k_{s - 1} - \rho} \right)\left( {k_{s} - k_{s - 1} + \rho} \right)}{12}} & {10.3{.2}D}\end{matrix}$

Note that these quantities are parameter free, even though thedistribution of U and the variance of C depend on probabilities of eachstate occurring. The variance in equation 10.3.2D is derived in appendix10C.

For the intrastate uniform model, the conditional variance of U can bewritten as for s=2, 3, . . . , S−1

$\begin{matrix}{{{Var}\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = \frac{\left( {k_{s} - k_{s - 1}} \right)^{2} - \rho^{2}}{12}} & {10.3{.2}E}\end{matrix}$

Thus, the variance is an increasing function of the distance between thestrikes. In an example embodiment, the auction sponsor can decrease thevariance of replication P&L, all other things being equal, by decreasingthe distance between the strikes. This result holds for the intrastateuniform model, but will hold for other example embodiments as well.

It is worth considering three special cases for this model. In the casewhere ρ=(k_(s)−k_(s−1))/2, there are two possible outcomes in state s soU is binomially distributed with the two values k_(s−1) andk_(s−1)+ρ=k_(s−1)+(k_(s)−k_(s−1))/2=(k_(s)+k_(s−1))/2. In this case, theconditional mean and the conditional variance is for s=2, 3, . . . , S−1

$\begin{matrix}\begin{matrix}{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = \frac{k_{s - 1} + k_{s} - \rho}{2}} \\{= \frac{k_{s - 1} + \left( {k_{s - 1} + {2\rho}} \right) - \rho}{2}} \\{= {k_{s - 1} + \frac{\rho}{2}}}\end{matrix} & {10.3{.2}F} \\\begin{matrix}{{{Var}\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = \frac{\left( {k_{s} - k_{s - 1} - \rho} \right)\left( {k_{s} - k_{s - 1} + \rho} \right)}{12}} \\{= \frac{\left( {{2\rho} - \rho} \right)\left( {{2\rho} + \rho} \right)}{12}} \\{= \frac{\rho^{2}}{4}}\end{matrix} & {10.3{.2}G}\end{matrix}$

In the special case of ρ=k_(s)−k_(s−1), the underlying only takes on thevalue k_(s−1) in the range of state s. Therefore, the conditional meanand the conditional variance is for s=2, 3, . . . , S−1

E[U|k _(s−1) ≦U<k _(s) ]=k _(s−1)  10.3.2H

Var[U|k_(s−1)≦U<k_(s)]=  10.3.2I

The case of ρ=0 implies that U is continuous, and in this case, for s=2,3, . . . , S−1

$\begin{matrix}{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = \frac{k_{s - 1} + k_{s}}{2}} & {10.3{.2}J} \\{{{Var}\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = \frac{\left( {k_{s} - k_{s - 1}} \right)^{2}}{12}} & {10.3{.2}K}\end{matrix}$

In contrast to the intrastate uniform model, another example embodimentsmight assume that the probability mass function of U is non-negative andtakes the form

Pr[U=u|k _(s−1) ≦U<k _(s)]=ρ(Γ_(s)+Φ_(s) u)  10.3.2L

This restriction allows the probability mass function to have a non-zeroslope intrastate, as opposed to the intrastate uniform model where theprobability mass function has a slope of zero intrastate. An exampleembodiment might estimate the parameters Γ_(s) and Φ_(s) of this modelsuch that these parameters minimize

(Pr _(Γ) _(s) _(,Φ) _(s) [k _(s−2) ≦U<k _(s−1) ]−p _(s−1))²+(Pr _(Γ)_(s) _(,Φ) _(s) [k _(s) ≦U<k _(s+1) ]−p _(s+1))²  10.3.2M

where Pr_(Γ) _(s) _(,Φ) _(s) [k _(s−2)≦U<k_(s−1)] denotes theprobability of state s−1 occurring based on Γ_(s) and Φ_(s), Pr_(Γ) _(s)_(,Φ) _(s) [k _(s)≦U<k_(s+1)] denotes the probability of state s+1occurring based on Γ_(s) and Φ_(s), and P_(s−1) and p_(s+1) denote theprobability that the state's s−1 and s+1 occur based on the auctionpricing.

Example

Consider the change in US nonfarm payrolls auction for October 2001 withthe strikes −425, −375, −325, −275, −225 and −175. In addition to theassumptions above for the intrastate uniform model, assume that

E[U|U<−425]=−450.50  10.3.2N

E[U|−175≦U]=−150.50  10.3.20

Table 10.3.2-1 shows Pr[k_(s−1)≦U<k_(s)], E[U|k_(s−1)≦U<k_(s)], andVar[U|k_(s−1)≦U<k_(s)] based on this intrastate uniform model. Theprobabilities of each state occurring are equal to those from table10.3.1-2 by assumption. Note that the conditional expectations and theconditional variance for the intrastate uniform model are different thanthose quantities for the discretized normal model in table 10.3.1-2.

TABLE 10.3.2-1 The probabilities, the conditional mean, and theconditional variance for the intrastate uniform model. State 2 State 3State 4 State 5 State 6 State 1 −425 <= U < −375 <= U < −325 <= U < −275<= U < −225 <= U < State 7 U < −425 −375 −325 −275 −225 −175 −175 <= UProbability of State 0.0342 0.1011 0.2162 0.2813 0.2225 0.1071 0.0375Conditional Expectation: −450.50 −400.50 −350.50 −300.50 −250.50 −200.50−150.50 E[U|state s] Conditional Variance: 208.25 208.25 208.25 208.25208.25 208.25 208.25 Var[U|state s]Table 10.3.2-2 shows the replicating digitals, the prices, and thevariance based on the intrastate uniform model. Note that thereplicating digitals for the discretized normal model and intrastateuniform model are the same for the digital call, the digital put, andthe range binary. Note that these replicating values are different forall other options.

TABLE 10.3.2-2 The replicating digitals, price of strategy, and varianceof different strategies based on the intrastate uniform model. State 2State 3 State 4 State 5 State 6 State 1 −425 <= U < −375 <= U < −325 <=U < −275 <= U < −225 <= U < State 7 Price of Replication DerivativeStrategy U < −425 −375 −325 −275 −225 −175 −175 <= U Strategy VarianceBuy a digital call struck 0.00 0.00 0.00 1.00 1.00 1.00 1.00 0.6484 0 at−325 Buy a digital put struck 1.00 1.00 1.00 1.00 0.00 0.00 0.00 0.63290 at −275 Buy a range binary with 0.00 0.00 1.00 1.00 1.00 0.00 0.000.7200 0 strikes of −375 and −225 Buy a vanilla call struck 0.00 0.000.00 24.50 74.50 124.50 174.50 43.3494 135.03 at −325 Buy a vanilla putstruck 175.50 125.50 75.50 25.50 0.00 0.00 0.00 42.1964 131.79 at −275Buy a call spread strikes 0.00 0.00 24.50 74.50 124.50 150.00 150.0075.6493 149.95 at −375 and −225 Buy a put spread strikes at 150.00150.00 125.50 75.50 25.50 0.00 0.00 74.3507 149.95 −375 and −225

FIGS. 27A, 27B, and 27C show the functions d and C for a vanilla calloption with a strike of −325 computed using the intrastate uniformmodel. FIGS. 28A, 28B, and 28C show the functions d and C for a callspread with strikes of −375 and −225 also using the intrastate uniformmodel.

10.4 Replication P&L for a Set of Orders

Previous sections showed how to compute the replication P&L for a singleorder for a specific derivatives strategy. This section shows how tocompute the replication P&L on a set of orders or an entire auction.

10.4.1 Replication P&L in the General Case

As before, assume U takes on values in Ω, where Ω has a countable numberof elements. Assume that the sample space Ω is divided into S disjointand non-empty subsets Ω₁, Ω₂, . . . , Ω_(S). Assume that Pr[U=u] is theprobability that outcome u occurs. Therefore,

$\begin{matrix}{{p_{s} = {{\sum\limits_{u\; ɛ\; \Omega_{s}}{{\Pr \left\lbrack {U = u} \right\rbrack}\mspace{14mu} {for}\mspace{14mu} s}} = 1}},2,\ldots \mspace{14mu},S} & {10.4{.1}A}\end{matrix}$

where p_(s) denotes the probability that state s occurs as defined inequation 10.1C.

Let J denote the number of filled customer orders and let these ordersbe indexed by the variable j,j=1, . . . , J. Let d_(j) denote the payoutfunction for the strategy for order j. For example if the jth order is acall spread with strikes k_(v) and k_(w), then

$\begin{matrix}{{d_{j}(U)} = \left\{ \begin{matrix}0 & {{{for}\mspace{14mu} U} < k_{v}} \\{U - k_{v}} & {{{for}\mspace{14mu} k_{v}} \leq U < k_{w}} \\{k_{w} - k_{v}} & {{{for}\mspace{14mu} k_{w}} \leq U}\end{matrix} \right.} & {10.4{.1}B}\end{matrix}$

Denote the filled notional payout amount for order j as x_(j). It isworth noting that the derivations in sections 10.1, 10.2, and 10.3implicitly assumed a notional payout value of 1 unit for each order. Letx denote the vector of length J, whose jth element is x_(j).

Let a_(j,s) denote the replicating digital for state s for order j. Forinstance, if the jth order is a call spread then the replicatingdigitals are

$\begin{matrix}{a_{j,s} = \left\{ \begin{matrix}0 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\{{E\left\lbrack U \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} - k_{v}} & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},w} \\{k_{w} - k_{v}} & {{{{for}\mspace{14mu} s} = {w + 1}},{w + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.4{.1}C}\end{matrix}$

Further let

$\begin{matrix}{{\underset{\_}{e}}_{j} = {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}{E\left\lbrack {d_{j}(U)} \middle| {U \in \Omega_{s}} \right\rbrack}}} & {10.4{.1}D}\end{matrix}$

Let C denote the replication P&L for this set of orders (in sections10.1, 10.2, and 10.3, C previously denoted the replication P&L for asingle order). The replication P&L for this set of orders if the ordersare buys of strategies d_(j) is

$\begin{matrix}{C = {\sum\limits_{j = 1}^{J}{x_{j}\left\lbrack {a_{j,s} - {d_{j}(U)} + {\underset{\_}{e}}_{j}} \right\rbrack}}} & {10.4{.1}E}\end{matrix}$

In this case, one can compute the expected replication P&L and thevariance of replication P&L from the auction as follows

$\begin{matrix}{{E\lbrack C\rbrack} = {\sum\limits_{u\; {ɛ\Omega}}{{\Pr \left\lbrack {U = u} \right\rbrack}{C(u)}}}} & {10.4{.1}F} \\{{{Var}\lbrack C\rbrack} = {\left( {\sum\limits_{u\; {ɛ\Omega}}{{\Pr \left\lbrack {U = u} \right\rbrack}{C(u)}^{2}}} \right) - \left( {E\lbrack C\rbrack} \right)^{2}}} & {10.4{.1}G}\end{matrix}$

(Note that C depends on the outcome u of U and equation 10.4.1F andequation 10.4.1 G makes that explicit by writing C(u)). Using formula10.4.1E one can compute the infimum replication P&L for the set of buyorders by computing the replication P&L over all possible values u of U.In the event that the sample space Ω takes on an uncountable number ofvalues, formulas 10.4.1F and 10.4.1G will require modification.

10.4.2 Replication P&L for Special Cases

-   -   Consider the following types of derivative strategies:    -   Digital calls, digital puts, and range binaries    -   Vanilla calls and vanilla puts    -   Call spreads and put spreads    -   Straddles and collared straddles    -   Forwards and collared forwards        These derivative strategies all have the property that their        payout functions d can be written as piece wise linear        functions. The section below derives formulas for the        replication variance for auctions with these derivative        strategies.

Let D be a matrix with J rows and S columns. Define the element in thejth row and sth column D_(j,s) as follows

$\begin{matrix}{D_{j,s} = \left\{ \begin{matrix}\begin{matrix}{{1\mspace{14mu} {if}\mspace{14mu} {the}\mspace{14mu} {replication}\mspace{14mu} {risk}\mspace{14mu} {for}}\mspace{14mu}} \\{{order}\mspace{11mu} j\mspace{14mu} {is}\mspace{14mu} {an}\mspace{14mu} {increasing}\mspace{14mu} {function}}\end{matrix} \\{{of}\mspace{14mu} U\mspace{14mu} {over}\mspace{14mu} {state}\mspace{14mu} s} \\{\mspace{14mu} \begin{matrix}{0\mspace{14mu} {if}\mspace{14mu} {the}\mspace{14mu} {replication}\mspace{14mu} {risk}\mspace{14mu} {for}} \\{{order}\mspace{14mu} j\mspace{14mu} {over}\mspace{14mu} {state}\mspace{14mu} s\mspace{14mu} {is}\mspace{14mu} {zero}}\end{matrix}} \\{\mspace{11mu} \begin{matrix}{{- 1}\mspace{14mu} {if}\mspace{14mu} {the}\mspace{14mu} {replication}\mspace{14mu} {risk}\mspace{14mu} {for}} \\{{order}\mspace{14mu} j\mspace{14mu} {is}\mspace{14mu} {an}\mspace{14mu} {decreasing}\mspace{14mu} {function}}\end{matrix}} \\{{of}\mspace{14mu} U\mspace{14mu} {over}\mspace{14mu} {state}\mspace{14mu} s}\end{matrix} \right.} & {10.4{.2}A}\end{matrix}$

Because digital calls, digital puts, and range binaries have noreplication P&L, then if order j is either a buy or sell of one of theseinstruments then

D_(j,s)=0 for s=1, 2, . . . , S  10.4.2B

If order j is a buy of a call spread with strikes k_(v) and k_(w) (or asell of a put spread with strikes k_(v) and k_(w)), then

$\begin{matrix}{D_{j,s} = \left\{ \begin{matrix}0 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\1 & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},w} \\0 & {{{{for}\mspace{14mu} s} = {w + 1}},{w + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.4{.2}C}\end{matrix}$

Similarly if order j is a sell of a call spread with strikes of k_(v)and k_(w) (or a buy of a put spread with strikes k_(v) and k_(w)) then

$\begin{matrix}{D_{j,s} = \left\{ \begin{matrix}0 & {{{{for}\mspace{14mu} s} = 1},2,\ldots \mspace{14mu},v} \\{- 1} & {{{{for}\mspace{14mu} s} = {v + 1}},{v + 2},\ldots \mspace{14mu},w} \\0 & {{{{for}\mspace{14mu} s} = {w + 1}},{w + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {10.4{.2}\; D}\end{matrix}$

Next, it is worth considering two special cases to compute the varianceof the replication P&L.

Case I: Var[U]<∞. In this case, one can compute the variance ofreplication P&L for an auction with the following strategies:

-   -   Digital calls, digital puts, and range binaries    -   Vanilla calls and vanilla puts    -   Call spreads and put spreads    -   Straddles and collared straddles    -   Forwards and collared forwards        Let U_(new) be a vector of length S defined such that the sth        element of U_(new) is

I[UεΩ_(s)](E[U|UεΩ_(s)]−U)  10.4.2E

for s=1, 2, . . . , S. Note, of course, that U_(new) does not depend onorder j. The replication P&L from an auction with these orders is

$\begin{matrix}\begin{matrix}{{{Var}\lbrack C\rbrack} = {{Var}\left\lbrack {x^{T} \times D \times U_{new}} \right\rbrack}} \\{= {x^{T} \times D \times {{Var}\left\lbrack U_{new} \right\rbrack} \times D^{T} \times x}}\end{matrix} & {10.4{.2}\; G}\end{matrix}$

Because of the definition of U_(new) and the fact that(E[U|k_(s−1)≦U<k_(s)]−U) is mean 0, then Var[U_(new)] is a diagonalmatrix where the element in the sth diagonal position isp_(s)Var[U|k_(s−1)≦U<k_(s)].

Case II: Var[U]=∞. In this case, the equations from Case I can bemodified to compute the variance of replication P&L for auctions withthe following instruments, which all have finite replication P&L (seetable 10.2.2-1):

-   -   Digital calls, digital puts, and range binaries    -   Call spreads and put spreads    -   Collared straddles    -   Collared forwards        Let U_(new) be a vector of length S defined such that the sth        element of U_(new) is

I[UεΩ_(s)](E[U|UεΩ_(s)]−U)  10.4.2H

for s=2, . . . , S−1 and let the first element and Sth element equal 0.The replication P&L from an auction with these orders is

$\begin{matrix}{{{Var}\lbrack C\rbrack} = {{Var}\left\lbrack {x^{T} \times D \times U_{new}} \right\rbrack}} & {10.4{.2}\; J} \\{\mspace{76mu} {= {x^{T} \times D \times {{Var}\left\lbrack U_{new} \right\rbrack} \times D^{T} \times x}}\;} & {10.4{.2}\; K}\end{matrix}$

Because of the definition of U_(new) and the fact that(E[U|k_(s−1)≦U<k_(s)]−U) is mean 0, then Var[U_(new)] is a diagonalmatrix where the element in the sth diagonal position isp_(s)Var[U|k_(s−1)≦U<k_(s)] for s=2, 3, . . . , S−1 and zero in element1 and S.

Example

To illustrate Case II, consider the example from section 10.3 with S=7states with strikes −425, −375, −325, −275, −225 and −175. Table10.4.2-1 shows the D's for a buy of a call spread with strikes −375 and−225 and a buy of a put spread with strikes −425 and −275, both withfilled notional amounts of 1. For this example, assume that theconditional variance of each state is modeled according to theintrastate uniform model of section 10.3.2 as shown in table 10.3.2-1.Table 10.4.2-1 shows that the variance of replication P&L for the callspread and put spread is 149.95 and 124.66 respectively. For J=2, thesetwo orders combined together in an auction have a replication varianceof 67.40. Because of the netting in the D's from these orders in states3 and 4, the replication variance for these combined orders is less thanthe sum of the replication variance of each order. (In fact, thereplication variance for these combined orders is less than thereplication variance of each order individually, because the ordersnetted together have replication risk on states with lowerprobabilities.) This netting phenomenon is likely to be a feature ofmany different sets of orders, keeping replication P&L growing less thanlinearly in J, the number of orders filled.

TABLE 10.4.2-1 The Matrix D and Replication P&L for Multiple OrdersState 1 State 2 State 3 State 4 State 5 State 6 State 7 −425 <= U < −375<= U < −325 <= U < −275 <= U < −225 <= U < Replication DerivativeStrategy U < −425 −375 −325 −275 −225 −175 −175 <= U Variance Buy a callspread strikes 0 0 1 1 1 0 0 149.95 at −375 and −225 Buy a put spreadstrikes 0 −1 −1 −1 0 0 0 124.66 at −425 and −275

Appendix 10A: Notation Used in Section 10

a_(s): a scalar representing the replicating digital for strategy d fors=1, 2, . . . , S;a_(ij): a scalar representing the replicating quantity of digitals forstate (i,j) for i=1, 2, . . . , S₁ and j=1, 2, . . . , S₂ when U istwo-dimensional;a_(j,s): a scalar representing the replicating digital for order j instate s for j=1, 2, . . . , J and s=1, 2, . . . , S;C: a one-dimensional random variable representing the replication P&L tothe auction sponsor;d: a function representing the payout on a derivatives strategy based onthe underlying U, also d(U);d_(j): a function representing the payout on a derivatives strategy fororder j;D: a matrix with J rows and S columns containing 1's, 0's, and −1's;e: a scalar representing the minimum conditional expected value of d(U)across states s for s=1, 2, . . . , S;ē: a scalar representing the maximum conditional expected value of d(U)across states s for s=1, 2, . . . , S;e _(j): a scalar representing the minimum conditional expected value ofd_(j)(U) for order j across states s for s=1, 2, . . . , S;E: the expectation operator;Exp: the exponential function raising the argument to the power of e;ƒ_(μ,σ): the density of a normally distribution random variable withmean μ and standard deviation σ;I: the indicator function;Inf: the infimum function;J: a scalar representing the number of customer orders in an auction;k₀, k₁, . . . , k_(S): scalar quantities representing strikes for thecase when U is one-dimensional;k₀ ¹, k₁ ¹, k₂ ¹, k₃ ¹, . . . , k_(s) ₁ ¹: scalar quantitiesrepresenting strikes for U₁ for the case when U is two-dimensional;k₀ ², k₁ ², k₂ ², k₃ ², . . . , k_(s) ₁ ²: scalar quantitiesrepresenting strikes for U₂ for the case when U is two-dimensional;k_(*): a scalar representing the target strike for an option order;N: the cumulative distribution function for the standard normal;P_(s): a scalar representing the probability that state s or Ω_(s) hasoccurred for s=1, 2, . . . , S;p_(*): a scalar representing the target price for an option order;p_(ij): a scalar representing the probability that state (i,j) hasoccurred for i=1, 2, . . . , S₁ and j=1, 2, . . . , S₂ when U istwo-dimensional;Pr: the probability operator;R: the rounding function, which discretizes a continuous distribution;s: a scalar used to index across the states;S: a scalar representing the number of states;S₁: a scalar representing the number of states for U₁ when U istwo-dimensional;S₂: a scalar representing the number of states for U₂ when U istwo-dimensional;U: a random variable representing the underlying;u: a possible outcome of U from the sample space ΩU₁ and U₂: one-dimensional random variables representing the first andsecond elements of U when U is two-dimensional;U_(new): a random vector of length S where the sth element isI[UεΩ_(s)](E[U|UεΩ_(s)]−U) for s=1, 2, . . . , S;Var: the variance operator;x: a vector of length J of filled notional amounts x_(j);x_(j): a scalar representing the filled notional amount of order j,j=1,2, . . . , J;Ω: a set of points representing the sample space of U;Ω₁, Ω₂, . . . , Ω_(S): subsets of the sample space Ω;ρ: a scalar representing the rounding parameter;

Appendix 10B: The General Replication Theorem

This appendix derives the formulas for the replicating digitals, theinfimum replication P&L, and the variance of replication P&L for buysand sells of derivatives strategies.

As a review of notation from section 10.1, recall that U denotes theunderlying. Let Ω denote the sample space of U and let Ω₁, Ω₂, . . . . ,Ω_(S) represent the different sets of outcomes of U. Let d represent thederivatives strategy and define

$\begin{matrix}{\underset{\_}{e} \equiv {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}{E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}}} & {10\; {B.A}} \\{\overset{\_}{e} \equiv {\max\limits_{{s = 1},2,\ldots \mspace{14mu},S}{E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}}} & {10\; {B.B}}\end{matrix}$

The derivation below requires d to satisfy the following restriction

0≦e<ē<∞  10B.O

Let (a₁, a₂, . . . , a_(S−1), a_(S)) represent the positions in thereplicating digitals, and let C denote the replication P&L, which isgiven by the formula

$\begin{matrix}{C = {\sum\limits_{s = 1}^{S}\; {{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}\left\lbrack {a_{s} - {d(U)} + \underset{\_}{e}} \right\rbrack}}} & {10\; {B.D}}\end{matrix}$

General Replication Theorem. If (a₁, a₂, . . . , a_(S−1), a_(S)) areselected to minimize. Var[C] subject to E[C]=0, then for a buy of d

a _(s) =E[d(U)|UεΩ _(s) ]−e for s=1, 2, . . . , S  10B.E

where d satisfies condition 10B.C. The infimum replication P&L for a buyof d is

$\begin{matrix}{{\inf \; C} = {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}\left\lbrack {\inf\limits_{U \in \Omega_{s}}\left( {{E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack} - {d(U)}} \right)} \right\rbrack}} & {10\; {B.F}}\end{matrix}$

Further, for a sale of the derivatives strategy d, the replicatingdigitals are given by the formula

a _(s) =ē−E[d(U)|UεΩ _(s) ]−e for s=1, 2, . . . , S  10B.G

The infimum replication P&L for a sell of d is given by

$\begin{matrix}{{\inf \; C} = {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}\left\lbrack {\inf\limits_{U \in \Omega_{s}}\left( {{d(U)} - {E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}} \right)} \right\rbrack}} & {10\; {B.H}}\end{matrix}$

The variance of replication P&L for both buys and sells of d is

$\begin{matrix}{{{Var}\lbrack C\rbrack} = {\sum\limits_{s = 1}^{S}\; {p_{s}{{Var}\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}}}} & {10\; {B.I}}\end{matrix}$

Proof. First, begin with the derivation of the result for a buy of d. Inthis case

$\begin{matrix}{{{{Var}\lbrack C\rbrack} \equiv {E\left\lbrack \left( {C - {E\lbrack C\rbrack}} \right)^{2} \right\rbrack}} = {E\left\lbrack C^{2} \right\rbrack}} & {10{B.J}}\end{matrix}$

where the first equality is the definition of variance and the secondequality follows from the constraint E[C]=0. Since

$\begin{matrix}{C = {\sum\limits_{s = 1}^{S}\; {{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}\left\lbrack {a_{s} - {d(U)} + \underset{\_}{e}} \right\rbrack}}} & {10{B.K}}\end{matrix}$

Therefore,

$\begin{matrix}\begin{matrix}{C^{2} = {\sum\limits_{t = 1}^{S}{\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}{I\left\lbrack {U \in \Omega_{t}} \right\rbrack}\left( {a_{s} - {d(U)} + \underset{\_}{e}} \right)\left( {a_{t} - {d(U)} + \underset{\_}{e}} \right)}}}} \\{= {\left( {\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}^{2}\left( {a_{s} - {d(U)} + \underset{\_}{e}} \right)^{2}}} \right) +}} \\{\left( {\sum\limits_{{t = 1},{t \neq s}}^{S}{\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}{I\left\lbrack {U \in \Omega_{t}} \right\rbrack}\left( {a_{s} - {d(U)} + \underset{\_}{e}} \right)\left( {a_{t} - {d(U)} + \underset{\_}{e}} \right)}}} \right)}\end{matrix} & {10{B.L}}\end{matrix}$

Note that in the second term on the RHS of equation 10B.L, the crossproduct terms contain the quantity

I[UεΩ_(s)]I[UεΩ_(t)] for t≠s  10B.M

Since Ω_(s) and Ω_(t) are mutually exclusive for t≠s, then

$\begin{matrix}\begin{matrix}{C^{2} = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}^{2}\left( {a_{s} - {d(U)} + \underset{\_}{e}} \right)^{2}}}} \\{= {\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}\left( {a_{s} - {d(U)} + \underset{\_}{e}} \right)^{2}}}}\end{matrix} & {10{B.O}}\end{matrix}$

where the last equation follows from the fact that squaring an indicatorfunction leaves it unchanged, i.e. I²=I. Therefore, taking expectationsof both sides of equation 10B.O gives

$\begin{matrix}{{{Var}\lbrack C\rbrack} = {\sum\limits_{s = 1}^{S}{E\left\lbrack {{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}\left( {a_{s} - {d(U)} + \underset{\_}{e}} \right)^{2}} \right\rbrack}}} & {10{B.P}}\end{matrix}$

Taking the derivative with respect to a_(s) for s=1, 2, . . . , S andsetting to zero gives the first order condition

E[I[UεΩ _(s)](a _(s) −d(U)+ e )]=0 for s=1, 2, . . . , S  10B.Q

or

E[I[UεΩ _(s) ]a _(s) ]−E[I[UεΩ _(s) ]d(U)]+E[I[UεΩ _(s) ]e]=0 for s=1,2, . . . , S  10B.R

which implies that

p _(s) a _(s) −p _(s) E[d(U)|UεΩ_(s) ]+p _(s) e=0 for s=1, 2, . . . ,S  10B.S

Factoring out p_(s) and solving for a_(s) implies that

a _(s) =E[d(U)|UεΩ _(s) ]−e for s=1, 2, . . . , S  10B.T

Next, one needs to check that E[C]=0 because that assumption was used inthe derivation above. Now,

$\begin{matrix}{C = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}\left\lbrack {a_{s} - {d(U)} + \underset{\_}{e}} \right\rbrack}}} & {10{B.U}}\end{matrix}$

Substituting equation 10B.T for a_(s) into the equation 10B.O gives

$\begin{matrix}{C = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {U \in \Omega_{s}} \right\rbrack}\left( {{E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack} - {d(U)}} \right)}}} & {10{B.V}}\end{matrix}$

Taking the expectations of both sides

$\begin{matrix}\begin{matrix}{{E(C)} = {E\left\lbrack {\sum\limits_{s = 1}^{S}\; {{I\left\lbrack {U\; {ɛ\Omega}_{s}} \right\rbrack}\left\lbrack {{E\left\lbrack {{d(U)}{U\; {ɛ\Omega}_{s}}} \right\rbrack} - {d(U)}} \right\rbrack}} \right\rbrack}} \\{= {\sum\limits_{s = 1}^{S}\; {E\left\lbrack {{I\left\lbrack {U\; {ɛ\Omega}_{s}} \right\rbrack}\left\lbrack {{E\left\lbrack {{d(U)}{U\; {ɛ\Omega}_{s}}} \right\rbrack} - {d(U)}} \right\rbrack} \right\rbrack}}} \\{= {\sum\limits_{s = 1}^{S}\; \left( {{p_{s}{E\left\lbrack {{d(U)}{U\; \in \Omega_{s}}} \right\rbrack}} - {p_{s}{E\left\lbrack {{d(U)}{U\; \in \Omega_{s}}} \right\rbrack}}} \right)}} \\{= 0}\end{matrix} & {10{B.W}}\end{matrix}$

To compute the variance of the replication P&L, recall equation 10B.P

$\begin{matrix}{{{Var}\lbrack C\rbrack} = {\sum\limits_{s = 1}^{S}\; {E\left\lbrack {{I\left\lbrack {U\; \in \Omega_{s}} \right\rbrack}\left\lbrack {a_{s} - {d(U)} + \underset{\_}{e}} \right\rbrack}^{2} \right\rbrack}}} & {10{B.X}}\end{matrix}$

Now

E[I[UεΩ _(s) ]a _(s) −d(U)+ e ]² ]=p _(s) E[(a _(s) −d(U)+ e )² |UεΩ_(s)]  10B.Y

Note that by definition of a_(s) in equation 10B.T

E[a _(s) −d(U)+ e|UεΩ _(s)]=0  10B.Z

Therefore,

$\begin{matrix}\begin{matrix}{{p_{s}{E\left\lbrack {\left( {a_{s} - {d(U)} + \underset{\_}{e}} \right)^{2}{U \in \Omega_{s}}} \right\rbrack}} = {p_{s}{{Var}\left\lbrack {{a_{s} - {d(U)} + \underset{\_}{e}}{U \in \Omega_{s}}} \right\rbrack}}} \\{= {p_{s}{{Var}\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}}}\end{matrix} & {10{B.{AA}}}\end{matrix}$

where the final equality follows from the fact that a₃ and e areconstants and don't impact the variance. Thus,

$\begin{matrix}{{{Var}\lbrack C\rbrack} = {\sum\limits_{s = 1}^{S}{p_{s}{{Var}\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack}}}} & {10{B.{AB}}}\end{matrix}$

Furthermore, the infimum replication P&L can be computed as follows

$\begin{matrix}{{\inf \; C} = {\min\limits_{{s = 1},2,\ldots \mspace{14mu},S}\left\lbrack {\inf\limits_{U \in \Omega_{s}}\left( {{E\left\lbrack {{d(U)}{U \in \Omega_{s}}} \right\rbrack} - {d(U)}} \right)} \right\rbrack}} & {10{B.{AC}}}\end{matrix}$

To distinguish replicating digitals for a buy of strategy d andreplicating digitals for a sell of strategy d, it is useful totemporarily use a_(s) to denote the replicating digitals for a buy ofstrategy d and ã_(s) to denote the replicating digital for a sell ofstrategy d. Outside of this discussion here, a_(s) denotes thereplicating digitals for both buys or sells of the derivatives strategyd.

A sell of strategy d can be handled by converting this sell into acomplementary buy order such that the combined replicating portfoliopays out the same amount regardless of what state occurs. In this case,denote the replicating digitals for the complementary buy as ã_(s) andthus

ã _(s) +a _(s)=constant for s=1, 2, . . . , S  10B.AD

The minimum such constant satisfying this equation and keeping ã_(s)non-negative is ē−e. Therefore,

ã _(s) +a _(s) =ē−e for s=1, 2, . . . , S  10B.AE

which implies that

ã _(s) =ē−e−a _(s) for s=1, 2, . . . , S  10B.AF

Since

a _(s) =E[d(U)|UεΩ_(s) ]−e for s=1, 2, . . . , S  10B.AG

Therefore,

ã _(s) =ē−E[d(U)|UεΩ_(s)] for s=1, 2, . . . , S  10B.AH

The formulas for the variance of replication P&L and the infimumreplication P&L for sells of d follow from equation 10B.AH.

Appendix 10C: Derivations from Section 10.3

This appendix derives results cited in section 10.3.1 and section10.3.2.

Derivation of Equation 10.3.1B from Section 10.3.1

This section derives equation 10.3.1B from the global normal model inSection 10.3.1. If U is normally distributed, then, the conditionalexpectation for s=2, 3, . . . , S−1 is given by

$\begin{matrix}{{E\left\lbrack {U{k_{s - 1} \leq U < k_{s}}} \right\rbrack} = \frac{\int_{k_{s - 1}}^{k_{s}}{{{uf}_{\mu,\sigma}(u)}\ {u}}}{\Pr \left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}} & {10{C.A}}\end{matrix}$

where ƒ_(μ,σ) denotes the normal density with mean μ and standarddeviation σ. Now,

$\begin{matrix}{{\Pr \left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack} = {{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}}} & {10{C.B}}\end{matrix}$

where N denotes the cumulative distribution function for the standardnormal. Further,

$\begin{matrix}{{\int_{k_{s - 1}}^{k_{s}}{{{uf}_{\mu,\sigma}(u)}\ {u}}} = {\int_{\frac{k_{s - 1} - \mu}{\sigma}}^{\frac{k_{s} - \mu}{\sigma}}{\left( {\mu + {\sigma \; z}} \right){f_{0,1}(z)}\ {z}}}} & {10{C.C}}\end{matrix}$

where Z=(U−μ)/σ. Therefore,

$\begin{matrix}\begin{matrix}{{E\left\lbrack {U{k_{s - 1} \leq U < k_{s}}} \right\rbrack} = \frac{\int_{k_{s - 1}}^{k_{s}}{{{uf}_{\mu,\sigma}(u)}\ {u}}}{{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}}} \\{= \frac{\int_{\frac{k_{s - 1} - \mu}{\sigma}}^{\frac{k_{s} - \mu}{\sigma}}{\left( {\mu + {\sigma \; z}} \right){f_{0,1}(z)}\ {z}}}{{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}}} \\{= {\frac{\mu {\int_{\frac{k_{s - 1} - \mu}{\sigma}}^{\frac{k_{s} - \mu}{\sigma}}{{f_{0,1}(z)}\ {z}}}}{{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}} +}} \\{\frac{\sigma {\int_{\frac{k_{s - 1} - \mu}{\sigma}}^{\frac{k_{s} - \mu}{\sigma}}{{{zf}_{0,1}(z)}\ {z}}}}{{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}}} \\{= {\mu + \frac{\sigma {\int_{\frac{k_{s - 1} - \mu}{\sigma}}^{\frac{k_{s} - \mu}{\sigma}}{{{zf}_{0,1}(z)}\ {z}}}}{{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}}}} \\{= {\mu + \frac{\sigma {\int_{\frac{k_{s - 1} - \mu}{\sigma}}^{\frac{k_{s} - \mu}{\sigma}}{\frac{z}{\sqrt{2\pi}}{\exp \left( {- \frac{z^{2}}{2}} \right)}}}}{{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}}}} \\{= {\mu - \frac{{\frac{\sigma}{\sqrt{2\pi}}{\exp \left( {- \frac{z^{2}}{2}} \right)}}_{\frac{k_{s - 1} - \mu}{\sigma}}^{\frac{k_{s} - \mu}{\sigma}}}{{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}}}} \\{= {\mu + \frac{\begin{bmatrix}{{\frac{\sigma}{\sqrt{2\pi}}{\exp \left( {- \frac{\left( {k_{s - 1} - \mu} \right)^{2}}{2\sigma^{2}}} \right)}} -} \\{\exp \left( {- \frac{\left( {k_{s} - \mu} \right)^{2}}{2\sigma^{2}}} \right)}\end{bmatrix}}{{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}}}}\end{matrix} & {10{C.D}}\end{matrix}$

Equation 10C.D matches equation 10.3.1B and so this concludes thederivation.Derivation of Equation 10.3.2D from Section 10.3.2

This section derives equation 10.3.2D, the variance for the intrastateuniform model. The derivation for the expected value is straightforwardand not presented.

Let the variable Z_(s) be defined as

$\begin{matrix}{Z_{s} = \frac{\left\lbrack {U{k_{s - 1} \leq U < k_{s}}} \right\rbrack - k_{s - 1}}{\rho}} & {10{C.E}}\end{matrix}$

The random variable [U|k_(s−1)≦U<k_(s)] takes on the values

k_(s−1),k_(s−1)+ρ,k_(s−1)+2ρ, . . . , k_(s)−ρ  10C.F

all with equal probability, since U is assumed to be uniformlydistributed intrastate. Therefore, Z_(s) takes on the values

0, 1, 2, . . . , (k_(s)−k_(s−1)−ρ)/ρ  10C.G

all with equal probability. An example of a random variable X taking onthe values 0, 1, 2, . . . , n−1 and n (all outcomes equally probable),described on page 141 in Evans, Hastings, and Peacock, StatisticalDistributions (Second Edition, Wiley Interscience, New York), has avariance

$\begin{matrix}{{{Var}(X)} = \frac{n\left( {n + 2} \right)}{12}} & {10{C.H}}\end{matrix}$

Thus, using this result with n=(k_(s)−k_(s−1)−ρ)/ρ implies that

$\begin{matrix}\begin{matrix}{{{Var}\left\lbrack Z_{s} \right\rbrack} = \frac{\left( {k_{s} - k_{s - 1} - \rho} \right)\left( {k_{s} - k_{s - 1} - \rho + {2\rho}} \right)}{12\rho^{2}}} \\{= \frac{\left( {k_{s} - k_{s - 1} - \rho} \right)\left( {k_{s} - k_{s - 1} + \rho} \right)}{12\rho^{2}}}\end{matrix} & {10{C.I}}\end{matrix}$

Therefore,

$\begin{matrix}\begin{matrix}{{{Var}\left\lbrack {U{k_{s - 1} \leq U < k_{s}}} \right\rbrack} = {\rho^{2}{{Var}\left\lbrack Z_{s} \right\rbrack}}} \\{= \frac{\left( {k_{s} - k_{s - 1} - \rho} \right)\left( {k_{s} - k_{s - 1} + \rho} \right)}{12}}\end{matrix} & {10{C.J}}\end{matrix}$

Equation 10C.J matches equation 10.3.2D and so this concludes thederivation.

11. REPLICATING AND PRICING DERIVATIVES STRATEGIES USING VANILLA OPTIONS

Financial market participants express market views and construct hedgesusing a number of contingent claims, such as derivatives strategies,including vanilla derivatives strategies (e.g. vanilla calls, vanillaputs, vanilla spreads, and vanilla straddles) and digital derivativesstrategies (e.g. digital calls, digital puts, and digital ranges). Usingthe techniques described in section 10, an auction sponsor can usedigital options to approximate or replicate these derivativesstrategies.

Replicating contingent claims, such as derivatives strategies usingdigital options exposes the auction sponsor to replication risk, therisk derived from synthesizing derivatives strategies for customersusing only digital options. To keep replication risk low, the auctionsponsor may only be able to offer customers the ability to tradederivatives strategies with low replication risk, which may includevanilla strategies with strikes that are close together. In fact,customers may demand vanilla strategies with a wider range of strikes,requiring the auction sponsor to take on higher replication risk. Tooffer the full range of strikes demanded by customers, the auctionsponsor may be exposed to a significant amount of replication risk whenusing digital options to replicate customer orders.

This section shows how an auction sponsor can eliminate replication riskby using vanilla options either alone, or together with digital options,instead of digital options alone, as the replicating claims establishedin the demand-based auction, to replicate digital and vanilladerivatives strategies in an example embodiment. This approach allowsthe auction sponsor to offer a wider range of strikes, which mayincrease customer demand in the auctions and better aggregate liquidity.This increased customer demand and liquidity will likely result inhigher fee income for the auction sponsor.

In an example embodiment, this replicating approximation may be amapping from parameters of, for example, vanilla options to the vanillareplicating basis. This mapping could be an automatic function builtinto a computer system accepting and processing orders in thedemand-based market or auction. The replicating approximation enablesauction participants or customers to interface with the demand-basedmarket or auction, side by side with customers who trade digitaloptions, notes and swaps, as well as other DBAR-enabled products withoutexposing the auction sponsor to replication risk. FIG. 29 shows thisvisually. All customer orders, including orders for both digital andvanilla options, are aggregated together into a single pool. Thisapproach can help increase the overall liquidity and risk pricingefficiency of the demand-based auction by increasing the variety andnumber of participants in the market or auction.

The remainder of section 11 proceeds as follows. Section 11.1 provides abrief review from Section 10, on how an auction sponsor can replicatederivatives strategies using digital replication claims (also referredto as replicating digital options) as the replicating claims for theauction (also referred to as replication claims). Next, section 11.2shows how an auction sponsor can replicate derivatives strategies usingvanilla replication claims (also referred to as replicating vanillaoptions). Section 11.3 extends the results from section 11.2 to considermore general cases. Section 11.4 develops the mathematical principlesfor computing the DBAR equilibrium. Section 11.5 discusses two examples,and section 11.6 concludes with a discussion of an augmented vanillareplicating basis.

11.1 Replicating Derivatives Strategies Using Digital Options

This section briefly reviews how an auction sponsor can replicatederivatives strategies using digital options (for a more detaileddiscussion, see section 10). Section 11.1.1 introduces the notation andset-up. Section 11.1.2 discusses the digital replicating claims, alsoreferred to as replicating digitals or replicating digital options.Section 11.1.3 shows how an auction sponsor can replicate digital andvanilla derivatives strategies based on these digital replicatingclaims. Section 11.1.4 computes the auction sponsor's replication P&L.

11.1.1 Notation and Set-Up

For simplicity, assume that the underlying U (also referred to as theevent or the underlying event) is one-dimensional. As in section 10, letρ denote the smallest measurable unit of U, or the level of precision towhich the underlying U is reported. For example, ρ equals 0.1 if theunderlying U is US CPI. In certain cases, ρ may be referred to as thetick size of the underlying.

Assume that the auction sponsor allows customers to trade derivativesstrategies with strikes k₁, k₂, . . . , k_(S−1), corresponding tomeasurements of the event U that are possible outcomes of U, such that

k ₁ <k ₂ <k ₃ < . . . <k _(S−2) <k _(S−1)  11.1.1A

Assume that the strikes k₁, k₂, . . . , k_(S−1) are all multiples of ρ.

Define k₀ as the lower bound of U, i.e. U is the largest value thatsatisfies

Pr[U<k₀]=0  11.1.1B

In the event that there is no such finite k₀ satisfying equation11.1.1B, let k₀=−∞. Define k_(S) as the upper bound, i.e. k_(S) is thesmallest value such that

Pr[U>k_(S)]=0  11.1.1C

In the event that there is no such finite k_(S) satisfying equation11.1.1C, set k_(S)=∞. Here, k₀ and k_(S) are not strikes that customerscan trade, but they will be useful mathematically in representingcertain equations below.

For derivatives strategies with a single strike, that strike willtypically be denoted by k_(v) where 1≦v≦S−1. For derivatives strategieswith two strikes, the lower strike will typically be denoted by k_(v)and the upper strike will typically be denoted by k_(w) where 1≦v<w≦S−1.

11.1.2 The Digital Replicating Claims

In an example embodiment, the auction sponsor may replicate derivativesstrategies using digital options. For example, the auction sponsor mayuse S such digital options (one more option than the number of strikes)for replication. For notation, let d^(s) denote the payout function,also referred to as the payout profile, on the sth such digitalreplicating claim for s=1, 2, . . . , S. The first digital replicatingclaim will be the digital put struck at k₁ which has a payout functionof

$\begin{matrix}{{d^{1}(U)} = \left\{ \begin{matrix}1 & {U < k_{1}} \\0 & {k_{1} \leq U}\end{matrix} \right.} & {11.1{.2}A}\end{matrix}$

The sth digital replicating claim for s=2, 3, . . . , S−1 is a digitalrange or range binary with strikes of k_(s−1) and k_(s), which has apayout function of

$\begin{matrix}{{d^{s}(U)} = \left\{ \begin{matrix}0 & {U < k_{s - 1}} \\1 & {k_{s - 1} \leq U < k_{s}} \\0 & {k_{s} \leq U}\end{matrix} \right.} & {11.1{.2}B}\end{matrix}$

The Sth digital replicating claim is a digital call struck at k_(S−1)with payout function

$\begin{matrix}{{d^{S}(U)} = \left\{ \begin{matrix}0 & {U < k_{S - 1}} \\1 & {k_{S - 1} \leq U}\end{matrix} \right.} & {11.1{.2}C}\end{matrix}$

FIG. 30 and table 11.1.2 display these digital replication claims. Thisset of claims is referred to as the digital replicating basis. Here,regardless of the outcome of the underlying, exactly one digitalreplicating claim expires in-the-money.

TABLE 11.1.2 The digital replicating claims in a DBAR auction. Range forClaim Non-Zero Number Payout Replicating Claim 1 U < k₁ Digital putstruck at k₁ 2 k₁ ≦ U < k₂ Digital range with strikes of k₁ and k₂ . . .. . . . . . s − 1 k_(s−2) ≦ U < k_(s−1) Digital range with strikes ofk_(s−2) and k_(s−1) s k_(s−1) ≦ U < k_(s) Digital range with strikes ofk_(s−1) and k_(s) s + 1 k_(s) ≦ U < k_(s+1) Digital range with strikesof k_(s) and k_(s+1) . . . . . . . . . S − 1 k_(S−2) ≦ U < k_(S−1)Digital range with strikes of k_(S−2) and k_(S−1) S k_(S−1) ≦ U Digitalcall struck at k_(S−1)

11.1.3 Replicating Derivatives Strategies with Digital ReplicationClaims

Let d denote the payout function or payout profile for a derivativesstrategy which is European style, i.e. its payout is based solely on thevalue of the underlying on expiration. Additionally, “derivativesstrategy d” in this specification refers to the payout function orpayout profile of the derivatives strategy, since a derivatives strategyis often identified by its payout function. Let a_(s) denote the amountor number of the sth digital replicating claim, also referred to as thereplication weight for this derivatives strategy d. The number or amountof each replicating claim is determined as a function of the payoutprofile or payout function d of the derivatives strategy, and the fullset of all the replicating claims that replicate or approximate thederivatives strategy can be referred to as the replication set for thederivatives strategy. In an example embodiment, the replicating weightsfor a buy of this derivatives strategy d are

a _(s) =E[d(U)|k_(s−1) ≦U<k _(s)] s=1, 2, . . . , S  11.1.3A

Here, the amount of the sth digital claim is the conditional expectedvalue of the payout of the derivatives strategy d, given that theunderlying U is greater than or equal to k_(s−1) and strictly less thank_(s). To compute this conditional expected value, the auction sponsormight assume for piecewise linear functions d that

$\begin{matrix}{{E\left\lbrack {d(U)} \middle| {k_{s - 1} \leq U < k_{s}} \right\rbrack} = {d\left( \frac{k_{s - 1} + k_{s} - \rho}{2} \right)}} & {11.1{.3}B}\end{matrix}$

Section 10.3.2 refers to equation 11.1.3B as the intrastate uniformmodel.

As now shown, the auction sponsor can use equations 11.1.3A and 11.1.3Bto compute the digital replicating weights (a₁, a₂, . . . , a_(S−1),a_(S)) for a digital range, a vanilla call spread, and a vanilla putspread to form replication sets for each of these derivativesstrategies. For the replication weights of additional derivativesstrategies using the digital replication basis, see section 10.2.

A digital range or range binary pays out a specified amount if, uponexpiration, the underlying U is greater than or equal to a lower strike,denoted by k_(v), and strictly less than a higher strike, denoted byk_(w). The payout function d for this digital range is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {U < k_{v}} \\1 & {k_{v} \leq U < k_{w}} \\0 & {k_{w} \leq U}\end{matrix} \right.} & {11.1{.3}C}\end{matrix}$

For a buy order of a digital range with strike prices of k_(v) and k_(w)the replicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{s = 1},2,\ldots \mspace{14mu},v} \\1 & {{s = {v + 1}},{v + 2},\ldots \mspace{14mu},w} \\0 & {{s = {w + 1}},{w + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {11.1{.3}D}\end{matrix}$

A buy of a vanilla call spread is the simultaneous buy of a vanilla callwith a lower strike k_(v) and the sell of a vanilla call with a higherstrike k_(w). The payout function d for this vanilla call spread is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {U < k_{v}} \\{U - k_{v}} & {k_{v} \leq U < k_{w}} \\{k_{w} - k_{v}} & {k_{w} \leq U}\end{matrix} \right.} & {11.1{.3}E}\end{matrix}$

For a buy order for a vanilla call spread with strikes of k_(v) andk_(w) the replicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{s = 1},2,\ldots \mspace{14mu},v} \\{\frac{k_{s - 1} + k_{s} - \rho}{2} - k_{v}} & {{s = {v + 1}},{v + 2},\ldots \mspace{14mu},w} \\{k_{w} - k_{v}} & {{s = {w + 1}},{w + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {11.1{.3}F}\end{matrix}$

based on the intrastate uniform model of equation 11.1.3B.

A buy of a vanilla put spread is the simultaneous buy of a vanilla putwith a higher strike k_(w) and the sell of a vanilla put with a lowerstrike k_(v). The payout function d for this vanilla put spread is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{k_{w} - k_{v}} & {U < k_{v}} \\{k_{w} - U} & {k_{v} \leq U < k_{w}} \\0 & {k_{w} \leq U}\end{matrix} \right.} & {11.1{.3}G}\end{matrix}$

For a buy order of a vanilla put spread with strikes of k_(w) and k_(v)the replicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{k_{w} - k_{v}} & {{s = 1},2,\ldots \mspace{14mu},v} \\{k_{w} - \frac{k_{s - 1} + k_{s} - \rho}{2}} & {{s = {v + 1}},{v + 2},\ldots \mspace{14mu},w} \\0 & {{s = {w + 1}},{w + 2},\ldots \mspace{14mu},S}\end{matrix} \right.} & {11.1{.3}H}\end{matrix}$

based on the intrastate uniform model of equation 11.1.3B.

11.1.4 Replication P&L

Let e(U) denote the payout on the replicating portfolio based on thereplication weights (a₁, a₂, a_(S−1), a_(S)) for strategy d. Note thate(U) can be written as

$\begin{matrix}{{{e(U)} \equiv {\sum\limits_{s = 1}^{S}{a_{s}{d^{s}(U)}}}} = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}a_{s}}}} & {11.1{.4}A}\end{matrix}$

where I denotes the indicator function, equaling one when its argumentis true and zero otherwise. Let C^(R)(U) denote the replication P&L tothe auction sponsor (note that section 10 uses the variable C to denotereplication P&L). If C^(R)(U) is positive (negative), then the auctionsponsor receives a profit (a loss) from the replication of the strategy.The replication P&L CR(U) is given by the following formula for a buyorder of the strategy d with a minimum payout of 0

$\begin{matrix}{{{C^{R}(U)} \equiv {{e(U)} - {d(U)}}} = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left\lbrack {a_{s} - {d(U)}} \right\rbrack}}} & {11.1{.4}B}\end{matrix}$

Note that for a digital range, equations 11.1.3C and 11.1.3D imply thatreplication P&L C^(R)(U) is zero regardless of the outcome of U.However, for each of a vanilla call spread and a vanilla put spread thereplication P&L C^(R)(U) will generally be non-zero. The replication P&Lfor a vanilla call spread is

$\begin{matrix}\begin{matrix}{{C^{R}(U)} = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left\lbrack {a_{s} - {d(U)}} \right\rbrack}}} \\{= {\sum\limits_{s = {v + 1}}^{w}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left\lbrack {\frac{k_{s - 1} + k_{s} - \rho}{2} - k_{v} - \left( {U - k_{v}} \right)} \right\rbrack}}} \\{= {\sum\limits_{s = {v + 1}}^{w}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left\lbrack {\frac{k_{s - 1} + k_{s} - \rho}{2} - U} \right\rbrack}}}\end{matrix} & {11.1{.4}C}\end{matrix}$

These results hold more generally. When using the digital replicationbasis, the auction sponsor replicates digital strategies with zeroreplication P&L, whereas the auction sponsor replicates vanilla optionswith non-zero replication P&L.

11.2 Replicating Claims Using a Vanilla Replicating Basis

This section discusses how to replicate derivatives strategies using avanilla replicating basis. Section 11.2.1 discusses the assumptionsbehind this framework. Section 11.2.2 defines the vanilla replicatingbasis. Section 11.2.3 presents the general replication theorem, in orderto form replication sets for any type of derivatives strategy as afunction of the payout profile or payout function d of the derivativesstrategy, the replication sets including one or more of the replicatingvanilla options, and sometimes a replicating digital option, as well.Using this theorem, section 11.2.4 shows how an auction sponsor canreplicate digital derivatives, and section 11.2.5 shows how an auctionsponsor can replicate vanilla derivatives.

11.2.1 Assumptions

This section discusses the five assumptions that will be used to derivethe general replication theorem. These assumptions will later be relaxedin section 11.3.

The first assumption regards the spacing of the strikes k₁, k₂, . . . ,k_(S−1) of the different derivatives strategies

-   -   Assumption 1: k_(s)−k_(s−1)≧2ρ s=2, 3, . . . , S−1

Assumption 1 requires the strikes to be set far enough apart such thatat least two outcomes are between adjacent strikes.

Assumptions 2 and 3 relate to the distribution of the underlying U.

-   -   Assumption 2: E[U²]<∞    -   Assumption 3: There do not exist a finite k₀ and k_(S) such that        Pr[U<k₀]=0 and Pr[U>k_(S)]=0.

Assumption 2 requires the second moment of U or equivalently thevariance of U to be finite. Assumption 3 requires that the underlying Uhas only unbounded support.

Assumptions 4 and 5 regard the payout on the derivatives strategy d.Assume that d takes on the following form

$\begin{matrix}{{d(U)} = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left( {\alpha_{s} + {\beta_{s}U}} \right)}}} & {{Assumption}\mspace{14mu} 4}\end{matrix}$

This assumption restricts d to be a piecewise linear function.

For notation, let d and d denote functions of the derivatives strategy dcomputed as follows

$\begin{matrix}{\overset{\_}{d} = {\max\left\lbrack {\underset{k_{1} \leq U < k_{S - 1}}{d(U)},{E\left\lbrack {d(U)} \middle| {U \geq k_{s - 1}} \right\rbrack},{E\left\lbrack {d(U)} \middle| {U < k_{1}} \right\rbrack}} \right\rbrack}} & {11.2{.1}A} \\{\underset{\_}{d} = {\min\left\lbrack {\underset{k_{1} \leq U < k_{S - 1}}{d(U)},{E\left\lbrack {d(U)} \middle| {U \geq k_{S - 1}} \right\rbrack},{E\left\lbrack {d(U)} \middle| {U < k_{1}} \right\rbrack}} \right\rbrack}} & {11.2{.1}B}\end{matrix}$

In many cases as shown below, d will be the maximum payout of thederivatives strategy d and d will be the minimum payout of thederivatives strategy d.

Assumption 5 is as follows

-   -   Assumption 5: d≦α_(s)+β_(s)k_(s) s=2, 3, . . . , S−1

This assumption ensures that the replication weights defined in section11.2.3 are non-negative. Strategies that satisfy assumptions 4 and 5include digital calls, digital puts, range binaries, vanilla calls,vanilla puts, vanilla call spreads, vanilla put spreads, vanillastraddles, collared vanilla straddles, forwards, and collared forwards.

11.2.2 The Vanilla Replicating Basis

This section introduces the vanilla replicating basis based on theassumptions in section 11.2.1. The vanilla replicating basis has a totalof 2S−2 replication claims, compared to S replicating claims for thedigital replicating basis. Note that the quantity 2S−2 is twice thenumber of defined strikes in a DBAR auction. The term “vanillareplicating basis” is something of a misnomer because two of thesereplicating claims are digital options.

Several of the replicating claims described below have a knockout orbarrier. All of these knockouts are European style, i.e. they are onlyin effect on expiration of the option, and thus do not depend on thepath of the underlying over the life of the option.

Let d^(s) denote the payout function for the vanilla replication claimsfor s=1, 2, . . . , 2S−2. The first vanilla replicating claim is adigital put with strike k₁ with payout function

$\begin{matrix}{{d^{1}(U)} = \left\{ \begin{matrix}1 & {U < k_{1}} \\0 & {k_{1} \leq U}\end{matrix} \right.} & {11.2{.2}A}\end{matrix}$

The second replicating claim has the following payout function

$\begin{matrix}{{d^{2}(U)} = \left\{ \begin{matrix}0 & {U < k_{1}} \\\frac{k_{2} - U}{k_{2} - k_{1}} & {k_{1} \leq U < k_{2}} \\0 & {k_{2} \leq U}\end{matrix} \right.} & {11.2{.2}B}\end{matrix}$

The payout of the second replicating claim is proportional to that of avanilla put struck at k₂ which has a European knockout below k₁. Notethat the second replicating claim has a payout of 1 at U=k₁. The thirdreplicating claim has a payout that is proportional to a vanilla callstruck at k₁ which has a European knock out at k₂. Mathematically,

$\begin{matrix}{{d^{3}(U)} = \left\{ \begin{matrix}0 & {U < k_{1}} \\\frac{U - k_{1}}{k_{2} - k_{1}} & {k_{1} \leq U < k_{2}} \\0 & {k_{2} \leq U}\end{matrix} \right.} & {11.2{.2}C}\end{matrix}$

In the general case for s=2, 3, . . . , S−1

$\begin{matrix}{{d^{{2s} - 2}(U)} = \left\{ \begin{matrix}0 & {U < k_{s - 1}} \\\frac{k_{s} - U}{k_{s} - k_{s - 1}} & {k_{s - 1} \leq U < k_{s}} \\0 & {k_{s} \leq U}\end{matrix} \right.} & {11.2{.2}D} \\{{d^{{2s} - 1}(U)}\left\{ \begin{matrix}0 & {U < k_{s - 1}} \\\frac{U - k_{s - 1}}{k_{s} - k_{s - 1}} & {k_{s - 1} \leq U < k_{s}} \\0 & {k_{s} \leq U}\end{matrix} \right.} & {11.2{.2}E}\end{matrix}$

For the s=2S−4 and s=2S−3 the replication claims are

$\begin{matrix}{{d^{{2S} - 4}(U)} = \left\{ \begin{matrix}0 & {U < k_{S - 2}} \\\frac{k_{S - 1} - U}{k_{S - 1} - k_{S - 2}} & {k_{S - 2} \leq U < k_{S - 1}} \\0 & {k_{S - 1} \leq U}\end{matrix} \right.} & {11.2{.2}F} \\{{d^{{2S} - 3}(U)}\left\{ \begin{matrix}0 & {U < k_{S - 2}} \\\frac{U - k_{S - 2}}{k_{S - 1} - k_{S - 2}} & {k_{S - 2} \leq U < k_{S - 1}} \\0 & {k_{S - 1} \leq U}\end{matrix} \right.} & {11.2{.2}G}\end{matrix}$

Note that the even-numbered replicating claims have a negatively slopedpayout between the strikes, similar to standard vanilla puts. Theodd-numbered replicating claims have a positively sloped payout betweenthe strikes similar to standard vanilla calls. The 2S−2^(nd) replicatingclaim is a digital call struck at k_(S−1).

$\begin{matrix}{{d^{{2S} - 2}(U)} = \left\{ \begin{matrix}0 & {U < k_{S - 1}} \\1 & {k_{S - 1} \leq U}\end{matrix} \right.} & {11.2{.2}H}\end{matrix}$

FIG. 31 and table 11.2.2 shows the vanilla replicating claims.

It is worth noting that the vanilla replicating claims are resealed insuch a way that for all U

$\begin{matrix}{{\sum\limits_{s = 1}^{{2S} - 2}{d^{s}(U)}} = 1} & {11.2{.2}I}\end{matrix}$

This feature will be used later in section 11.4.3, where the sum of theprices of the replicating claims are required to sum to one.

TABLE 11.2.2 The payout ranges and replicating claims for the vanillareplicating basis. Claim Payout European Number Range VanillaReplicating Claim Knockout? 1 U < k₁ Digital put struck at k₁ None 2 k₁≦ U < k₂ Rescaled vanilla put struck at k₂ Knockout at k₁ − ρ 3 k₁ ≦ U <k₂ Rescaled vanilla call struck at k₁ Knockout at k₂ 4 k₂ ≦ U < k₃Rescaled vanilla put struck at k₃ Knockout at k₂ − ρ 5 k₂ ≦ U < k₃Rescaled vanilla call struck at k₂ Knockout at k₃ . . . . . . . . . . .. 2s − 2 k_(s−1) ≦ U < k_(s) Rescaled vanilla put struck at k_(s)Knockout at k_(s−1) − ρ 2s − 1 k_(s−1) ≦ U < k_(s) Rescaled vanilla callstruck at k_(s−1) Knockout at k_(s) 2s k_(s) ≦ U < k_(s+1) Rescaledvanilla put struck at k_(s+1) Knockout at k_(s) − ρ 2s + 1 k_(s) ≦ U <k_(s+1) Rescaled vanilla call struck at k_(s) Knockout at k_(s+1) . . .. . . . . . . . . 2S − 4 k_(S−2) ≦ U < k_(S−1) Rescaled vanilla putstruck at k_(S−1) Knockout at k_(S−2) − ρ 2S − 3 k_(S−2) ≦ U < k_(S−1)Rescaled vanilla call struck at k_(S−2) Knockout at k_(S−1) 2S − 2k_(S−1) ≦ U Digital call struck at k_(S−1) NoneAs seen in the table above, it is worth noting that with the vanillareplication basis, two such claims often payout if the underlying is ina specific range. This distinguishes this basis from the digitalreplicating basis discussed in section 11.1 where only one replicatingclaim pays out regardless of the outcome of U.

11.2.3 General Replication Theorem for Buys and Sells of Digital andVanilla Derivatives

The following theorem shows how to construct the weights on the vanillareplicating portfolio denoted as (a₁, a₂, . . . , a_(2S−3), a_(2S−2)) ofstrategies d that satisfy the assumptions above.

General Replication Theorem. Under assumptions 1, 2, 3, 4, and 5, thevariance minimizing replicating weights for a buy of strategy d are

a ₁=α₁+β₁ E[U|U<k ₁ ]−d   11.2.3A

a _(2s−2)=α_(s)+β_(s) k _(s−1) −d s=2, 3, . . . , S−1  11.2.3B

a _(2s−1)=α_(s)+β_(s) k _(s) −d s=2, 3, . . . , S−1  11.2.3C

a _(2S−2)=α_(S)+β_(S) E[U|U≧k _(S−1) ]d   11.2.3D

For a sell of strategy d, the variance minimizing weights are

a ₁ = d−α ₁−β₁ E[U|U≧k ₁]  11.2.3E

a _(2s−1) = d−α ₁−β_(s) k _(s−1) s=2, 3, . . . , S−1  11.2.3F

a _(2s−2) = d−α ₁−β_(s) k _(s−1) s=2, 3, . . . , S−1  11.2.3G

a _(2S−2) = d−α ₁−β_(S) E[U|U≧k _(S−1)]  11.2.3H

The replication P&L C^(R)(U) for a buy of d is

C ^(R)(U)=β₁(U−E[U|U<k ₁])I[U<k ₁]+β_(S)(E[U|U k _(S−1) ]−U)I[U≧k_(S−1)]  11.2.3I

The replication P&L C^(R)(U) for a sell of d is

C ^(R)(U)=β₁(E[U|U<k ₁ ]−U)I[U<k ₁]+β_(S)(U−E[U|U≧k _(S−1)])I[U≧k_(S−1)]  11.2.3J

Proof of General Replication Theorem: See Appendix 11A.

Appendix 11A shows that the replication weights for a buy of d or a sellof d satisfy

min(a ₁ ,a ₂ , . . . , a _(2S−3) , a _(2S−2))=0  11.2.3K

Since the a's are non-negative, this ensures that aggregated customerpayouts (the y's defined in section 11.4.5) are also non-negative.

Note that the payout on the replicating portfolio for a buy of d plusthe payout on the replicating portfolio for a sell of d equals d=d forall values of U. Thus the payout on the replicating portfolio with a buyof d and a sell of d is constant and so, as expected, the portfolio isrisk free.

Consider the special case where as defined in assumption 4, d satisfies

β₁=β_(s)=0  11.2.3L

In this case, the payout of the derivatives strategy is constant andequal to α₁ if the underlying U is less than k₁. Similarly, the payoutof the derivatives strategy is constant and equal to α_(S) if theunderlying U is greater than or equal to k_(S−1). Under 11.2.3L, thereplicating portfolio is not only minimum variance but also hasreplication P&L C^(R)(U)=0 for every outcome U. This result applies todigital calls, digital puts, range binaries, vanilla call spreads,vanilla put spreads, collared vanilla straddles, and collared forwards.The vanilla replicating basis replicates these instruments with zeroreplication P&L.

The next section applies the general replication theorem to compute thevanilla replicating weights for different digital derivatives strategiesd.

11.2.4 Using the General Replication Theorem to Compute ReplicationWeights for Digital Options

To apply the general replication theorem above for digital derivativesstrategies, note that for the digital options discussed below d=1 andd=0. Further, digital options have the following parameters restrictions

β₁=β₂= . . . =β_(S)=0  11.2.4A

In addition, α_(s) will equal zero or one for s=1, 2, . . . , S.

The payout function d for a digital call with a strike price of k_(v) is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {U < k_{v}} \\1 & {k_{v} \leq U}\end{matrix} \right.} & {11.2{.4}B}\end{matrix}$

For a buy order of a digital call with a strike price of k_(v) thereplicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\1 & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.4}C}\end{matrix}$

For a sell order of a digital call with a strike price of k_(v) thereplicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}1 & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\0 & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.4}D}\end{matrix}$

A digital put pays out a specific quantity if the underlying is strictlybelow the strike, denoted k_(v), on expiration, and therefore its payoutfunction d is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}1 & {U < k_{v}} \\0 & {k_{v} \leq U}\end{matrix} \right.} & {11.2{.4}E}\end{matrix}$

For a buy order of a digital put with a strike price of k_(v) thereplicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}1 & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\0 & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.4}F}\end{matrix}$

For a sell order of a digital put with a strike price of k_(v) thereplicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\1 & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.4}G}\end{matrix}$

As defined in section 11.1.3, the payout function for a digital rangewith strikes k_(v) and k_(w) can be represented as

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {U < k_{v}} \\1 & {k_{v} \leq U < k_{w}} \\0 & {k_{w} \leq U}\end{matrix} \right.} & {11.2{.4}H}\end{matrix}$

For a buy order of a digital range with strikes k_(v) and k_(w) thereplicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\1 & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2w} - 1}} \\0 & {{s = {2w}},{{2w} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.4}I}\end{matrix}$

One can contrast equation 11.2.4I to equation 11.1.3D, which shows thereplicating weights for a digital range with the digital replicatingbasis. For a sell order of a digital range with strikes k_(v) and k_(w)the replicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}1 & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\0 & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2w} - 1}} \\1 & {{s = {2w}},{{2w} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.4}J}\end{matrix}$

11.2.5 Using the General Replication Theorem to Compute ReplicationWeights for Vanilla Derivatives

This section uses the general replication theorem to compute vanillareplication weights for vanilla derivatives. Note that for vanilladerivatives, β_(s) equals 0 or 1 for s=1, 2, . . . , S. In addition, forall the vanilla derivatives strategies described below, excludingforwards and collared forwards d=0. For notation, let the functionint[x] denote the greatest integer less than or equal to x.

Replicating Vanilla Call Options and Vanilla Put Options

The payout function d for a vanilla call with strike k_(v) is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {U < k_{v}} \\{U - k_{v}} & {k_{v} \leq U}\end{matrix} \right.} & {11.2{.5}A}\end{matrix}$

Note that in this case, d=0 and d=E[U|U≧k_(S−1)]−k_(v). For a buy orderfor a vanilla call with strike k_(v) the replication weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\{k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} - k_{v}} & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 3}} \\{{E\left\lbrack {U{U \geq k_{S - 1}}} \right\rbrack} - k_{v}} & {s = {{2S} - 2}}\end{matrix} \right.} & {11.2{.5}B}\end{matrix}$

For a sell order for a vanilla call with strike k_(v) the replicationweights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{E\left\lbrack {U{U \geq k_{S - 1}}} \right\rbrack} - k_{v}} & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\{{E\left\lbrack {U{U \geq k_{S - 1}}} \right\rbrack} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 3}} \\0 & {s = {{2S} - 2}}\end{matrix} \right.} & {11.2{.5}C}\end{matrix}$

The payout function d for a vanilla put with strike k_(v) is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{k_{v} - U} & {U < k_{v}} \\0 & {k_{v} \leq U}\end{matrix} \right.} & {11.2{.5}D}\end{matrix}$

Note that in this case, d=0 and d=k_(v)−E[U|U<k₁]. For a buy order for avanilla put with strike k_(v) the replication weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{k_{v} - {E\left\lbrack {U{U < k_{1}}} \right\rbrack}} & {s = 1} \\{k_{v} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = 2},3,\ldots \mspace{14mu},{{2v} - 1}} \\0 & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.5}E}\end{matrix}$

For a sell order for a vanilla put with strike k_(v) the replicationweights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {s = 1} \\{k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} - {E\left\lbrack {U{U < k_{1}}} \right\rbrack}} & {{s = 2},3,\ldots \mspace{14mu},{{2v} - 1}} \\{k_{v} - {E\left\lbrack {U{U < k_{1}}} \right\rbrack}} & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.5}F}\end{matrix}$

Replicating Vanilla Call Spreads and Vanilla Put Spreads

As discussed in section 11.1.3, the payout function d for a vanilla callspread with strikes k_(v) and k_(w) is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {U < k_{v}} \\{U - k_{v}} & {k_{v} \leq U < k_{w}} \\{k_{w} - k_{v}} & {k_{w} \leq U}\end{matrix} \right.} & {11.2{.5}G}\end{matrix}$

For this strategy, note that d=0 and d=k_(w)−k_(v). For a buy order fora vanilla call spread with strikes of k_(v) and k_(w) the replicatingweights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\{k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} - k_{v}} & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2w} - 1}} \\{k_{w} - k_{v}} & {{s = {2w}},{{2w} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.5}H}\end{matrix}$

One can contrast equation 11.2.5H to equation 11.1.3F, which shows thereplicating weights for a vanilla call spread with the digitalreplicating basis. For a sell order for a vanilla call spread withstrikes of k_(v) and k_(w) the replicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{k_{w} - k_{v}} & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\{k_{w} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2w} - 1}} \\0 & {{s = {2w}},{{2w} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.5}I}\end{matrix}$

Again, as discussed in section 11.1.3, the payout function d for a buyof a vanilla put spread with strikes k_(w) and k_(v) is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{k_{w} - k_{v}} & {U < k_{v}} \\{k_{w} - U} & {k_{v} \leq U < k_{w}} \\0 & {k_{w} \leq U}\end{matrix} \right.} & {11.2{.5}J}\end{matrix}$

For a vanilla put spread, note that d=0 and d=k_(w)−k_(v), which areidentical values for d to d for a vanilla call spread. For a buy orderfor a vanilla put spread with strikes of k_(w) and k_(v) the replicatingweights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{k_{w} - k_{v}} & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\{k_{w} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2w} - 1}} \\0 & {{s = {2w}},{{2w} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.5}K}\end{matrix}$

One can contrast equation 11.2.5K to equation 11.1.3H, which shows thereplicating weights for a vanilla put spread with the digitalreplicating basis. For a sell order for a vanilla put spread withstrikes of k_(w) and k_(v) the replicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{s = 1},2,\ldots \mspace{14mu},{{2v} - 1}} \\{k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} - k_{v}} & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2w} - 1}} \\{k_{w} - k_{v}} & {{s = {2w}},{{2w} + 1},\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} \right.} & {11.2{.5}L}\end{matrix}$

Replicating Vanilla Straddles and Collared Vanilla Straddles

A buy of a vanilla straddle is the simultaneous buy of a vanilla calland a vanilla put both with identical strike prices. A buy of a vanillastraddle is a bullish volatility strategy, in that the purchaser profitsif the outcome is very low or very high.

For a vanilla straddle with a strike of k_(v), the payout function d is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{k_{v} - U} & {U < k_{v}} \\{U - k_{v}} & {k_{v} \leq U}\end{matrix} \right.} & {11.2{.5}M}\end{matrix}$

Note that for this strategy d=max(E[U|U≧k_(S−1)]−k_(v), k_(v)−E[U|U<k₁])and d=0. Therefore, for a buy of a vanilla straddle

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{k_{v} - {E\left\lbrack {U{U < k_{1}}} \right\rbrack}} & {s = 1} \\{k_{v} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = 2},3,\ldots \mspace{14mu},{{2v} - 1}} \\{k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} - k_{v}} & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 3}} \\{{E\left\lbrack {U{U \geq k_{S - 1}}} \right\rbrack} - k_{v}} & {s = {{2S} - 2}}\end{matrix} \right.} & {11.2{.5}N}\end{matrix}$

For a sell of a vanilla straddle the replicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{\overset{\_}{d} - k_{v} + {E\left\lbrack {U{U < k_{1}}} \right\rbrack}} & {s = 1} \\{\overset{\_}{d} - k_{v} + k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = 2},3,\ldots \mspace{14mu},{{2v} - 1}} \\{\overset{\_}{d} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} + k_{v}} & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 3}} \\{\overset{\_}{d} - {E\left\lbrack {U{U \geq k_{S - 1}}} \right\rbrack} + k_{v}} & {s = {{2S} - 2}}\end{matrix} \right.} & {11.2{.5}O}\end{matrix}$

To avoid taking on replication risk, the auction sponsor may offerparticipants the ability to instead trade a collared vanilla straddlewhose payout can be written as

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{k_{v} - k_{1}} & {U < k_{1}} \\{k_{v} - U} & {k_{1} \leq U < k_{v}} \\{U - k_{v}} & {k_{v} \leq U < k_{S - 1}} \\{k_{S - 1} - k_{v}} & {k_{S - 1} \leq U}\end{matrix} \right.} & {11.2{.5}P}\end{matrix}$

Note that d=max[k_(S−1)−k_(v), k_(v)−k₁] for a collared vanilla straddleand d=0.For a buy order of a collared vanilla straddle with strike k_(v), thereplicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{k_{v} - k_{1}} & {s = 1} \\{k_{v} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = 2},3,\ldots \mspace{14mu},{{2v} - 1}} \\k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} & {{s = {2v}},{{2v} + 1},\ldots \mspace{14mu},{{2S} - 3}} \\{k_{S - 1} - k_{v}} & {s = {{2S} - 2}}\end{matrix} \right.} & {11.2{.5}Q}\end{matrix}$

Therefore, for a sell order of a vanilla straddle with strike k_(v) thereplicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{\max \left\lbrack {{k_{S - 1} - k_{v}},{k_{v} - k_{1}}} \right\rbrack} - k_{v} + k_{1}} & {s = 1} \\{{\max \left\lbrack {{k_{S - 1} - k_{v}},{k_{v} - k_{1}}} \right\rbrack} - k_{v} + k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = 2},3,\ldots \mspace{14mu},{{2v} - 1}} \\{{\max \left\lbrack {{k_{S - 1} - k_{v}},{k_{v} - k_{1}}} \right\rbrack} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} + k_{v}} & \begin{matrix}{{s = {2v}},{{2v} + 1},} \\{\ldots \mspace{14mu},{{2S} - 3}}\end{matrix} \\{{\max \left\lbrack {{k_{S - 1} - k_{v}},{k_{v} - k_{1}}} \right\rbrack} - k_{S - 1} + k_{v}} & {s = {{2S} - 2}}\end{matrix} \right.} & {11.2{.5}R}\end{matrix}$

Replicating Forwards and Collared Forwards

A forward pays out based on the underlying as follows

d(U)=U−π ^(ƒ)  11.2.5S

where π^(ƒ) denotes the forward price. Note that for a forward,d=E[U|U≧k_(S−1)]−π^(ƒ)and d=E[U|U<k₁]−π^(ƒ). In this case, for a buy ofa forward

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {s = 1} \\{k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} - {E\left\lbrack {U{U < k_{1}}} \right\rbrack}} & {{s = 2},3,\ldots \mspace{14mu},{{2S} - 3}} \\{{E\left\lbrack {U{U \geq k_{S - 1}}} \right\rbrack} - {E\left\lbrack {U{U < k_{1}}} \right\rbrack}} & {s = {{2S} - 2}}\end{matrix} \right.} & {11.2{.5}T}\end{matrix}$

For a sell of a forward

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{{E\left\lbrack {U{U \geq k_{S - 1}}} \right\rbrack} - {E\left\lbrack {U{U < k_{1}}} \right\rbrack}} & {s = 1} \\{{E\left\lbrack {U{U \geq k_{S - 1}}} \right\rbrack} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = 2},3,\ldots \mspace{14mu},{{2S} - 3}} \\0 & {s = {{2S} - 2}}\end{matrix} \right.} & {11.2{.5}U}\end{matrix}$

To avoid taking on replication P&L, the auction sponsor may offer acollared forward strategy with maximum and minimum payouts. Let π^(cf)denote the price on the collared forward. For a buy order of a collaredforward, the payout function d is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{k_{1} - \pi^{cf}} & {U < k_{1}} \\{U - \pi^{cf}} & {k_{1} \leq U < k_{S - 1}} \\{k_{S - 1} - \pi^{cf}} & {k_{S - 1} \leq U}\end{matrix} \right.} & {11.2{.5}V}\end{matrix}$

In this case, note that d=k_(S−1)−π^(cf) and d=k₁−π^(cf). For a buy of acollared forward the replicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {s = 1} \\{k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} - k_{1}} & {{s = 2},3,\ldots \mspace{14mu},{{2S} - 3}} \\{k_{S - 1} - k_{1}} & {s = {{2S} - 2}}\end{matrix} \right.} & {11.2{.5}W}\end{matrix}$

For a sell of a collared forward the replicating weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{k_{S - 1} - k_{1}} & {s = 1} \\{k_{S - 1} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = 2},3,\ldots \mspace{14mu},{{2S} - 3}} \\0 & {s = {{2S} - 2}}\end{matrix} \right.} & {11.2{.5}X}\end{matrix}$

It is worth noting that there is possibly infinite replication risk forvanilla calls and vanilla puts, vanilla straddles, and forwards.

Estimating the Conditional Expectation of the Underlying

Note that for buys and sells of vanilla calls, vanilla puts, vanillastraddles, and forwards, some of the replicating weights depend uponE[U|U<k₁] and E[U|U≧k_(S−1)]. These two quantities could be estimated,for example, using a non parametric approach based on a historical datasample on the underlying as follows. The auction sponsor could use theaverage of the observations below k₁ to estimate E[U|U<k₁], and theauction sponsor could use the average of the observations greater thanor equal to k_(S−1) to estimate E[U|U≧k_(S−1)]. Alternatively, theauction sponsor could estimate these quantities parametrically assumingthat U follows a certain distribution. To select the appropriatedistribution for U, the auction sponsor might employ techniques fromSection 10.3.1 in the subsections “Classes of Distributions for theUnderlying,” and “Selecting the Appropriate Distribution.”

For the case that U is normally distributed, recall equation 10.3.1Bfrom section 10

$\begin{matrix}{{E\left\lbrack {U{k_{s - 1} \leq U < k_{s}}} \right\rbrack} = {\mu + \frac{\frac{\sigma}{\sqrt{2\pi}}\left\lbrack {{\exp \left( {- \frac{\left( {k_{s - 1} - \mu} \right)^{2}}{2\sigma^{2}}} \right)} - {\exp \left( {- \frac{\left( {k_{s} - \mu} \right)^{2}}{2\sigma^{2}}} \right)}} \right\rbrack}{{N\left\lbrack \frac{k_{s} - \mu}{\sigma} \right\rbrack} - {N\left\lbrack \frac{k_{s - 1} - \mu}{\sigma} \right\rbrack}}}} & {11.2{.5}Y}\end{matrix}$

where π denotes the constant 3.14159 . . . , where N denotes thecumulative normal distribution, and where “exp” denotes the exponentialfunction or raising the argument to the power of e. Letting k_(s−1)→−∞and setting k_(s) equal to k₁, then equation 11.2.5Y simplifies to

$\begin{matrix}{{E\left\lbrack {U{U < k_{1}}} \right\rbrack} = {\mu - \frac{\frac{\sigma}{\sqrt{2\pi}}\left\lbrack {\exp \left( {- \frac{\left( {k_{1} - \mu} \right)^{2}}{2\sigma^{2}}} \right)} \right\rbrack}{N\left\lbrack \frac{k_{1} - \mu}{\sigma} \right\rbrack}}} & {11.2{.5}Z}\end{matrix}$

Letting k_(s)→∞ and setting equal to k_(S−1), then equation 11.2.5Ysimplifies to

$\begin{matrix}{{E\left\lbrack {U{k_{S - 1} \leq U}} \right\rbrack} = {\mu + \frac{\frac{\sigma}{\sqrt{2\pi}}\left\lbrack {\exp \left( {- \frac{\left( {k_{S - 1} - \mu} \right)^{2}}{2\sigma^{2}}} \right)} \right\rbrack}{1 - {N\left\lbrack \frac{k_{S - 1} - \mu}{\sigma} \right\rbrack}}}} & {11.2{.5}{AA}}\end{matrix}$

To estimate the parameters of the distribution of U such as μ and σabove, the auction sponsor might employ techniques discussed in Section10.3.1 in the subsection “Estimating the Parameters of theDistribution.”

11.3 Extensions to the General Replication Theorem

Section 11.2.1 discusses the five assumptions used to derive the generalreplication theorem. This section discusses how relaxing theseassumptions impacts the theorem.

11.3.1 Relaxing Assumption 1 on Strike Spacing

Assumption 1 is

k _(s) −k _(s−1)≧2ρ s=2, 3, . . . , S−1  11.3.1A

What happens when strikes are closer together and this assumption isviolated while assumptions 2, 3, 4, and 5 still hold?

Remember that an example embodiment may assume that the strikes aremultiples of the tick size ρ. Consider the special case where 11.3.1Aholds for all values of s except s=2 and

k ₂ =k ₁+ρ  11.3.1B

In this case, the first vanilla replicating claim is the digital putstruck at k₁ which is the first replicating claim from section 11.2.2.The second vanilla replicating claim is the rescaled European vanillaput struck at k₂ that knocks out at k₁−ρ. Note that since k₁ and k₂ arespaced ρ apart as dictated by 11.3.1B, this replicating claim pays out 1at k₁ and zero otherwise. Therefore, the second vanilla replicatingclaim is equivalent to a digital range with strikes of k₁ and k₂. Thethird vanilla replicating claim from section 11.2.2 is a rescaledEuropean vanilla call struck at k₁ that knocks out at k₂. Note that,under 11.3.1B, this instrument pays out zero for all outcomes of U andso can be eliminated from the replication basis in this case. Thus,under 11.3.1B, vanilla replicating claims two and three from section11.2.2 combine into a single digital range. Under 11.3.1B, the thirdreplicating claim for this setup corresponds to the fourth replicatingclaim from section 11.2.2. Under 11.3.1B, the fourth replicating claimcorresponds to the fifth replicating claim from section 11.2.2. And soon. Except for the second and third replication claims, all otherreplication claims under 11.3.1B are the same as those listed in table11.2.2. Under 11.3.1B, the number of replicating claims decreases from2S−2 to 2S−3. Table 11.3.1 shows the replicating claims for this case.

TABLE 11.3.1 The payout ranges, replicating claims in a DBAR auction forthe vanilla replicating basis with strikes satisfying equation 11.3.1B.Claim Payout European Number Range Vanilla Replicating Claim Knockout? 1U < k₁ Digital put struck at k₁ None 2 k₁ ≦ U < k₂ Digital range withstrikes at k₁ and k₂ None 3 k₂ ≦ U < k₃ Rescaled vanilla put struck atk₃ Knockout at k₂ − ρ 4 k₂ ≦ U < k₃ Rescaled vanilla call struck at k₂Knockout at k₃ . . . . . . . . . . . . 2s − 3 k_(s−1) ≦ U < k_(s)Rescaled vanilla put struck at k_(s) Knockout at k_(s−1) − ρ 2s − 2k_(s−1) ≦ U < k_(s) Rescaled vanilla call struck at k_(s−1) Knockout atk_(s) 2s − 1 k_(s) ≦ U < k_(s+1) Rescaled vanilla put struck at k_(s+1)Knockout at k_(s) − ρ 2s k_(s) ≦ U < k_(s+1) Rescaled vanilla callstruck at k_(s) Knockout at k_(s+1) . . . . . . . . . . . . 2S − 5k_(S−2) ≦ U < k_(S−1) Rescaled vanilla put struck at k_(S−1) Knockout atk_(S−2) − ρ 2S − 4 k_(S−2) ≦ U < k_(S−1) Rescaled vanilla call struck atk_(S−2) Knockout at k_(S−1) 2S − 3 k_(S−1) ≦ U Digital call struck atk_(S−1) NoneNext consider the case

k _(s) =k _(s−1) +ρ s=2, 3, . . . , S−1  11.3.1C

Here, all the strikes in the auction are spaced ρ apart. Once again, thesecond and third vanilla replicating claims of table 11.2.2 combine intoa single digital range. Similarly, the fourth and fifth replicatingclaims of table 11.2.2 also combine into a single digital range. And soon. For case 11.3.1C, there are S replicating claims and all thereplicating claims are digital options. These replicating claims are thedigital replicating claims listed in table 11.1.2.

Consider the more general case where

k _(s) =k _(s−1)+ρ  11.3.1D

for at least one value of s, 2≦s≦S−1. Here, the rescaled Europeanvanilla put struck at k_(s) that knocks out at k_(s−1)−ρ and theresealed vanilla call struck at k_(s−1) that knocks out at k_(s) combineto form one replicating instrument, a digital range with strikes ofk_(s−1) and k_(s). In this case, the total number of replicating claimsis equal to

$\begin{matrix}{2 + {\sum\limits_{s = 2}^{S - 1}{\min \left\lbrack {\frac{k_{s} - k_{s - 1}}{\rho},2} \right\rbrack}}} & {11.3{.1}E}\end{matrix}$

The two replicating claims that payout across strikes spaced ρ apartreduce to a single digital range.

11.3.2 Relaxing Assumption 2, the Finite Second Moment Assumption

Assumption 2 is

E[U²]<∞  11.3.2A

This assumption is required for computing the replication weights a₁ anda_(2S−2) for buys of d and for sells of d. Without this assumption,there is no minimum variance estimator over the range U<k₁ or U≧k_(S−1).

To relax assumption 2 with assumptions 1, 3, 4, and 5 still satisfied,consider first the case that

E[|U|]=∞  11.3.2B

Here, the expected value of the absolute value of U is infinite. In thiscase, the auction sponsor might choose replication weights a₁ anda_(2S−2) such that

Pr[a ₁−(d(U)− d )<0]=Pr[a ₁−(d(U)− d )>0]  11.3.2C

Pr[a _(2S−2)−(d(U)− d )<0]=Pr[a _(2S−2)−(d(U)− d )>0]  11.3.2D

In using these replication weights, the auction sponsor has an equalchance of a replication profit or a replication loss for a buy ofstrategy d. The auction sponsor can construct the replication weightsfor a sell of strategy d in a similar manner.

Next consider the case where

E[|U|]<∞ and E[U ²]=∞  11.3.2E

In this case the auction sponsor may use the replication weights fromthe general replication theorem as specified in 11.2.3A, 11.2.3D,11.2.3E, and 11.2.3H, though these replication weights are not minimumvariance weights since the variance is infinite. As an alternative, theauction sponsor may choose to use replication weights that satisfyequations 11.3.2C and 11.3.2D.

11.3.3 Relaxing Assumption 3, the Unbounded Assumption

Recall that the general replication theorem was derived under theassumption that underlying U is unbounded, i.e. there do not existfinite k₀ and k_(S) such that Pr[U<k₀]=0 and Pr[U>k_(S)]=0. Thisassumption is now relaxed in case 1 and case 2 below. The analysis belowrequires assumptions 1, 2, 4, and 5 to hold.

Case 1: The Underlying U is Bounded on One Side

Consider the case where the underlying U is bounded below, and where thelower bound is computed in equation 11.1.1B and denoted as k₀. Severalfinancial underlyings are bounded below with k₀=0, including

-   -   the prices of currencies in units of another currency;    -   the prices of commodities;    -   the prices of fixed income instruments;    -   the prices of equities; and    -   weather derivatives based on heating degree days and cooling        degree days over a set period.

In addition, several economic variables are measured in percentagechange terms (including the monthly percentage change in retail sales,the monthly change in average hourly earnings, or the quarterly changein GDP), and these variables are bounded below at k₀=−100%.

It is worth dividing case 1 into three sub-cases, case 1A, case 1B, andcase 1C.

Case 1A: k₁≦k₀. The auction sponsor may find it unnecessary to set thelowest (or first) strike k₁ at or below k₀, since in this case theprobability of the underlying U being at or below k₁ is zero. Consider,for example, the price of an equity where the lower bound k₀ equals 0.Financial exchanges generally do not offer customers put or call optionson a stock with a strike price of k₁=−1.

Case 1B: k₁=k₀+ρ. In this case, the replicating instruments are the same2S−2 replicating instruments as described in section 11.2.2. Further,note the replication formulas 11.2.3A-11.2.3H apply. For a₁, equation11.2.3A simplifies to

$\begin{matrix}\begin{matrix}{a_{1} = {{E\left\lbrack {{d(U)}{U < k_{1}}} \right\rbrack} - \underset{\_}{d}}} \\{= {{d\left( k_{0} \right)} - \underset{\_}{d}}}\end{matrix} & {11.3{.3}A}\end{matrix}$

Note that the value of a₁ does not depend on any parameters of thedistribution of U. In this case, a strategy d that satisfies assumption4 with β_(S)=0, for example a vanilla put, can be replicated using the2S−2 instruments with zero replication error.

Case 1C. k₁>k₀+ρ. In this case, there are a total of 2S−1^(st)replicating claims. The first two replicating instruments are a resealedEuropean vanilla put struck at k₁ with a European knockout out below k₀and a resealed European vanilla call struck at k₀ with a Europeanknockout at k₁. The third through 2S−1^(st) replication claimscorrespond to the second through 2S−2^(nd) replication claims from table11.2.2, respectively. These replication instruments, also referred to asreplication claims or replicating claims for the demand-based or DBARauction, are displayed in table 11.3.3A.

TABLE 11.3.3A The payout ranges, replicating claims in a DBAR auctionfor the vanilla replicating basis under case 1C. Claim Payout EuropeanNumber Range Vanilla Replicating Claim Knockout? 1 k₀ ≦ U < k₁ Rescaledvanilla put struck at k₁ Knockout at k₀ − ρ 2 k₀ ≦ U < k₁ Rescaledvanilla call struck at k₀ Knockout at k₁ 3 k₁ ≦ U < k₂ Rescaled vanillaput struck at k₂ Knockout at k₁ − ρ 4 k₁≦ U < k₂ Rescaled vanilla callstruck at k₁ Knockout at k₂ 5 k₂ ≦ U < k₃ Rescaled vanilla put struck atk₃ Knockout at k₂ − ρ 6 k₂ ≦ U < k₃ Rescaled vanilla call struck at k₂Knockout at k₃ . . . . . . . . . . . . 2s − 1 k_(s−1) ≦ U < k_(s)Rescaled vanilla put struck at k_(s) Knockout at k_(s−1) − ρ 2sk_(s−1 ≦ U < k) _(s) Rescaled vanilla call struck at k_(s−1) Knockout atk_(s) 2s + 1 k_(s) ≦ U < k_(s+1) Rescaled vanilla put struck at k_(s+1)Knockout at k_(s) − ρ 2s + 2 k_(s) ≦ U < k_(s+1) Rescaled vanilla callstruck at k_(s) Knockout at k_(s+1) . . . . . . . . . . . . 2S − 3k_(S−2) ≦ U < k_(S−1) Rescaled vanilla put struck at k_(S−1) Knockout atk_(S−2) − ρ 2S − 2 k_(S−2) ≦ U < k_(S−1) Rescaled vanilla call struck atk_(S−2) Knockout at k_(S−1) 2S − 1 k_(S−1) ≦ U Digital call struck atk_(S−1) None

In this case, the replicating weights for the first and second claimsare given by

a ₁=α₁+β₁ k ₀ −d   11.3.3B

a ₂=α₁+β₁ k ₁ −d   11.3.3C

Replication weights for the other replication claims follow the generalreplication theorem of section 11.2.3.

This discussion has focused on the case when U is bounded below andunbounded above. The case when U is bounded above and unbounded belowfollows a similar approach.

Case 2: The Underlying U is Bounded Both Below and Above

Several events of economic significance are bounded both below and aboveincluding

-   -   the change in a futures contract over a pre-specified time        period, where the futures contract can move a maximum number of        points (or ticks) up or down per day;    -   mortgage CPR rates, which are bounded between 0 and 1200;    -   diffusion indices such as German IFO and US ISM, which are        bounded between 0 and 100; and    -   economic variables that measure a percentage (not a percentage        change) such as the percentage of the work force unemployed,        which are bounded between 0% and 100%.        In this case, let k₀ be the lower bound as defined in equation        11.1.1B and assume that k₁=k₀+ρ (case 1B).

Case 2A: k_(S)≦k_(S−1). In this case, the maximum value of U is lessthan or equal to the maximum strike or equivalently, there is noprobability mass above the highest (or last) strike established in theauction. This may be an unlikely scenario as the auction sponsor may setstrikes over only the range of likely outcomes of U.

Case 2B: k_(s)>k_(S−1). In this case, there are 2S replicating claims.The first 2S−2 replicating claims are the first 2S−2 claims as listed intable 11.3.3A. The final two replicating claims are the rescaled vanillaput struck at k_(S) with a knockout at k_(S−1)×ρ, and a rescaled vanillacall struck at k_(S−1) with a knockout at k_(S). These replicationclaims are displayed in table 11.3.3B. In this case, all instruments canbe replicated with zero replication P&L.

TABLE 11.3.3B The payout ranges, replicating claims in a DBAR auctionfor the vanilla replicating basis for case 2B. Claim Payout EuropeanNumber Range Vanilla Replicating Claim Knockout? 1 k₀ ≦ U < k₁ Rescaledvanilla put struck at k₁ Knockout at k₀ − ρ 2 k₀ ≦ U < k₁ Rescaledvanilla call struck at k₀ Knockout at k₁ 3 k₁ ≦ U < k₂ Rescaled vanillaput struck at k₂ Knockout at k₁ − ρ 4 k₁ ≦ U < k₂ Rescaled vanilla callstruck at k₁ Knockout at k₂ 5 k₂ ≦ U < k₃ Rescaled vanilla put struck atk₃ Knockout at k₂ − ρ 6 k₂ ≦ U < k₃ Rescaled vanilla call struck at k₂Knockout at k₃ . . . . . . . . . . . . 2s − 1 k_(s−1) ≦ U < k_(s)Rescaled vanilla put struck at k_(s) Knockout at k_(s−1) − ρ 2sk_(s−1 ≦ U < k) _(s) Rescaled vanilla call struck at k_(s−1) Knockout atk_(s) 2s + 1 k_(s ≦) U < k_(s+1) Rescaled vanilla put struck at k_(s+1)Knockout at k_(s) − ρ 2s + 2 k_(s) ≦ U < k_(s+1) Rescaled vanilla callstruck at k_(s) Knockout at k_(s+1) . . . . . . . . . . . . 2S − 3k_(S−2) ≦ U < k_(S−1) Rescaled vanilla put struck at k_(S−1) Knockout atk_(S−2) − ρ 2S − 2 k_(S−2) ≦ U < k_(S−1) Rescaled vanilla call struck atk_(S−2) Knockout at k_(S−1) 2S − 1 k_(S−1) ≦ U Rescaled vanilla putstruck at k_(S) Knockout at k_(S−1) − ρ 2S k_(S−1) ≦ U Rescaled vanillacall struck at k_(S−1) Knockout at k_(S)

11.3.4 Relaxing Assumption 4, the Piecewise Linear Assumption

Note that the general replication theorem is derived using Assumption 4,the piecewise linear assumption

$\begin{matrix}{{d(U)} = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left( {\alpha_{s} + {\beta_{s}U}} \right)}}} & {11.3{.4}A}\end{matrix}$

In fact, some derivatives strategies d may not satisfy this equation.Instead, d may be, for example, a quadratic function of the underlyingor piecewise linear over more than S such pieces.

If assumption 4 is violated but assumptions 1, 2, 3, and 5 hold, thenthe auction sponsor might determine the replication weights (a₁, a₂, . .. , a_(2S−3), a_(2S−2)) for a buy of the strategy d using an ordinaryleast squares regression or OLS as follows. Select G possible outcomesof the underlying U denoted as (u₁, u₂, . . . , u_(G)), where G may belarge relative to S. Define the variables

y _(g) ^(OLS) =d(u _(g))− d g=1, 2, . . . , G  11.3.4B

y _(g,s) ^(OLS) =d ^(s)(u _(g))g=1, 2, . . . , G and s=1, 2, . . . ,2S−2  11.3.4C

Then, use the model

$\begin{matrix}{{y_{g}^{OLS} = {{\left( {\sum\limits_{s = 1}^{{2S} - 2}{a_{s}x_{g,s}^{OLS}}} \right) + {ɛ_{g}\mspace{14mu} g}} = 1}},2,\ldots \mspace{14mu},G} & {11.3{.4}D}\end{matrix}$

In equation 11.3.4D, they variable of the regression (the dependentvariable) is the value of d(U)−d over the specified range, and the xvariables of the regression (the independent variables) are the payoutsof the vanilla replication instruments (also referred to as thereplicating vanilla options). The regression slope coefficients are thevalues of (a₁, a₂, a_(2S−3), a_(2S−2)), which the auction sponsor canestimate by OLS. Once the replication weights are constructed for a buyof strategy d, then the auction sponsor can set a_(s) for a sell ofstrategy d equal to ( d−d) minus the replication weight a_(s) for a buyof strategy d.

Instead of using OLS to construct the replication weights, the auctionsponsor might construct (a₁, a₂, a_(2S−3), a_(2S−2)) using weightedleast squares, where the weight on the residual ε_(g) is proportional tothe probability that U=u_(g). Alternatively, the auction sponsor may runa regression constraining the coefficients to be non-negative.

11.3.5 Relaxing Assumption 5, the Regularity Condition on d

Assumption 5 is as follows

d≦α _(s)+β_(s) k _(s) ≦ d s=2, 3, . . . , S−1  11.3.5A

What happens when Assumption 5 is violated, while assumptions 1, 2, 3,and 4 hold?

Define the variables d′ and d′ as follows

d′=max( k,α ₂+β₂ k ₂,α₃+β₃ k ₃, . . . , α_(S−1)+β_(S−1) k_(S−1)  11.3.5B

d′=min( k,α ₂+β₂ k ₂,α₃+β₃ k ₃, . . . , α_(S−1)+β_(S−1) k_(S−1)  11.3.5C

In this case, the general replication theorem holds with equations11.2.3A-11.2.3H modified by replacing d with d′ and by replacing d withd′.

11.4 Mathematical Restrictions for the Equilibrium

The previous sections show how to replicate derivatives strategies usingthe vanilla replicating basis. This section discusses the mathematicalrestrictions for pricing and filling orders in a DBAR equilibrium.Section 11.4 in some cases draws from material in section 7. Thus, table11.4 shows some changes in notation between this section and thenotation in section 7.

TABLE 11.4 Notation differences between section 7 and section 11.Variable in Variable in Section 7 Section 11 Meaning of Variable T MTotal cleared premium m S Number of strikes plus one k_(i) θ_(s) Openingorder amount n J Number of customer orders

11.4.1 Opening Orders

In an example embodiment, the auction sponsor may enter initialinvestment amounts for each of the 2S−2 vanilla replicating claims,referred to as opening orders. Let the opening order premium be denotedas 0, for replicating claims s=1, 2, . . . , 2S−2. An example embodimentmay require

θ_(s)=0 s=1, 2, . . . , 2S−2  11.4.1A

Opening orders ensure that the DBAR equilibrium prices are unique. See,for example, the unique price equilibrium proof in section 7.11.

Let Θ be the total amount of opening orders computed as

$\begin{matrix}{\Theta = {\sum\limits_{s = 1}^{{2S} - 2}\theta_{s}}} & {11.4{.1}B}\end{matrix}$

The auction sponsor may determine the total amount of opening orders Θbased on a desired level of initial liquidity for the DBAR auction orbased on the desired level of computational efficiency for theequilibrium.

Once Θ is determined, the auction sponsor can use a variety of ways todetermine the individual opening order amounts θ₁, θ₂, . . . , θ_(2S−2)based on the auction sponsor's objective.

-   -   Maximize Expected Profit. The auction sponsor may wish to        maximize the expected profit from the opening orders. In this        case, the auction sponsor may make O_(s) proportional to the        auction sponsor's estimate of the fair value of the sth        replicating claim. Here, the auction sponsor sets

θ_(s) =Θ×E[d ^(s)(U)] s=1, 2, . . . , 2S−2  11.4.1C

-   -   The auction sponsor may compute this expected value using a        non-parametric approach or by assuming a specific distribution        for U. To select the appropriate distribution for U,

the auction sponsor might employ techniques from Section 10.3.1 in thesubsections “Classes of Distributions for the Underlying,” and“Selecting the Appropriate Distribution.”

-   -   Minimize Standard Deviation. The auction sponsor may wish to        minimize the standard deviation of opening order P&L. In this        case, the auction sponsor may enter the opening orders        proportional to the auction sponsor's estimate of the likely        final equilibrium price of the replicating claim. In this case,        let p_(s) denote the equilibrium price of the sth replicating        claim for s=1, 2, . . . , 2S−2. Then the auction sponsor sets

θ_(s) =Θ×E[p _(s) ] s=1, 2, . . . , 2S−2  11.4.1D

-   -   In this case, the expectation is taken over the auction        sponsor's estimate of the likely values of the final equilibrium        prices of the replicating claims.    -   Maximize the Minimum P&L. Alternatively, the auction sponsor may        choose to maximize the minimum P&L from the opening orders. In        this case, the auction sponsor sets

$\begin{matrix}{{\theta_{s} = \frac{\Theta}{{2S} - 2}}{{s = 1},2,\ldots \mspace{14mu},{{2S} - 2}}} & {11.4{.1}E}\end{matrix}$

-   -   Here, opening orders are equal for all replicating claims.

11.4.2 Customer Orders

Customers can submit orders to buy or sell derivatives strategiesfollowing standard derivative market protocols in a DBAR auction. Fornotation, assume that customers submit a total of J orders in theauction, indexed by j=1, 2, . . . , J. When submitting an order, thecustomer requests a specific number of contracts, denoted by r_(j). Fordigital options, the auction sponsor may adopt the convention that onecontract pays out $1 if the digital option expires in-the-money. Forvanilla derivatives, the auction sponsor may adopt the convention thatone contract pays out $1 per point that the option expires in-the-money.

Let d_(j) denote the payout function for the jth customer order. Similarto equation 11.2.1A and equation 11.2.1B, let

$\begin{matrix}{\overset{\_}{d_{j}} = {\max\left\lbrack {\underset{k_{1} \leq U < k_{S - 1}}{d_{j}(U)},{E\left\lbrack {d_{j}(U)} \middle| {U \geq k_{S - 1}} \right\rbrack},{E\left\lbrack {d_{j}(U)} \middle| {U < k_{1}} \right\rbrack}} \right\rbrack}} & {11.4{.2}A} \\{\underset{\_}{d_{j}} = {\min\left\lbrack {\underset{k_{1} \leq U < k_{S - 1}}{d_{j}(U)},{E\left\lbrack {d_{j}(U)} \middle| {U \geq k_{S - 1}} \right\rbrack},{E\left\lbrack {d_{j}(U)} \middle| {U < k_{1}} \right\rbrack}} \right\rbrack}} & {11.4{.2}B}\end{matrix}$

for the jth customer order, j=1, 2, . . . , J.

In a DBAR auction, customers may specify a limit price for each order.The limit price for a buy of a derivatives strategy represents themaximum price the customer is willing to pay for the derivativesstrategy specified. The limit price for a sell of a derivatives strategyrepresents the minimum price at which the customer is willing to sellthe derivatives strategy. For notation, let w_(j) denote the limit pricefor customer order j,

11.4.3 Pricing Derivative Strategies Based on the Prices of theReplicating Claims

Mathematically, the auction sponsor may require that

$\begin{matrix}{{p_{s} > 0}{s = 1},2,\ldots \mspace{14mu},{{2S} - 2}} & {11.4{.3}A} \\{{\sum\limits_{s = 1}^{{2S} - 2}p_{s}} = 1} & {11.4{.3}B}\end{matrix}$

Here, the auction sponsor requires that the prices of the vanillareplicating claims are positive and sum to one.

Based on these prices, the auction sponsor may determine the equilibriumprice of each derivatives strategy using the prices of the replicatingclaims as follows. Let π_(j) denote the equilibrium mid-price for thederivatives strategy requested in order j. For simplicity of exposition,assume here that the auction sponsor does not charge fees (see section7.8 for a discussion of fees). Let a_(j,s) denote the replication weightfor the sth replicating claim for a customer order computed using thegeneral replication theorem of section 11.2.3. Then, for a derivativesstrategy with payout function d_(j)

$\begin{matrix}{\pi_{j} \equiv {\sum\limits_{s = 1}^{{2S} - 2}{a_{j,s}p_{s}}}} & {11.4{.3}C}\end{matrix}$

Each strategy is priced as the sum of the product of the strategy'sreplicating weights and the prices of the respective vanilla replicatingclaims.

11.4.4 Adjusted Limit Prices and Determining Fills in a DBAR Auction

Following the discussion in section 7.8, let w_(j) ^(a) denoted theadjusted limit price for customer order j as follows. If order j is abuy of strategy d_(j), then

w _(j) ^(a) =w _(j)− d _(j)   11.4.4A

Similar to the replication weights a_(j,s) for a buy (see equations11.2.3A, 11.2.3B, 11.2.3C, and 11.2.3D), the adjusted limit price w_(j)^(a) is a function of d_(j) . For a sell of strategy d_(j),

w _(j) ^(a)= d _(j) −w _(j)  11.4.4B

Similar to the replication weights a_(j,s) for a sell (see equations11.2.3E, 11.2.3F, 11.2.3G, and 11.2.3H), the adjusted limit price w_(j)^(a) is a function of d_(j) .

In an example embodiment, the auction sponsor may determine fills in theauction based on the adjusted limit price as follows. Let x_(j) denotethe equilibrium number of filled contracts for order j for j=1, 2, . . ., J. When customer order j is a buy order, if the customer's adjustedlimit price w_(j) ^(a) is below the DBAR equilibrium price π_(j), thenthe customer's bid is below the market, and the customer's orderreceives no fill, so x_(j)=0. If the customer's adjusted limit pricew_(j) ^(a) is exactly equal to the DBAR equilibrium price then thecustomer's bid is at the market, and the customer's order may receive afill, so 0≦x_(j)≦r_(j). If the customer's adjusted limit price w_(j)^(a) is above the DBAR equilibrium price π_(j), then the customer's bidis above the market, and the customer's order is fully filled, sox_(j)=r_(j). Mathematically, the logic for a buy order or a sell orderis as follows

w_(j) ^(a)<π_(j)→x_(j)=0

w_(j) ^(a)=π_(j)→0≦x_(j)≦r_(j)

w_(j) ^(a)>π_(j)→x_(j)=r_(j)  11.4.4C

Note that in an example embodiment the equilibrium price of order j, isnot necessarily equal to w_(j) ^(a), the customer's adjusted limitprice. In an example embodiment, every buy order with an adjusted limitprice at or above the equilibrium price may be filled at thatequilibrium price. In an example embodiment, every sell order with anadjusted limit price at or below the equilibrium price may be filled atthat equilibrium price.

11.4.5 Equilibrium Pricing Conditions and Self-Hedging

Let M denote the total replicated premium paid in the auction computedas follows

$\begin{matrix}{M \equiv {\left( {\sum\limits_{j = 1}^{J}{x_{j}\pi_{j}}} \right) + {\sum\limits_{s = 1}^{{2S} - 2}\theta_{s}}}} & {11.4{.5}A}\end{matrix}$

Next, note that a_(j,s)x_(j) is the amount of replicating claim s usedto replicate order j. Define y_(s) as

$\begin{matrix}{y_{s} \equiv {\sum\limits_{j = 1}^{J}{a_{j,s}x_{j}}}} & {11.4{.5}B}\end{matrix}$

for s=1, 2, . . . , 2S−2. Here, y_(s) is the aggregate filled amountacross all customer orders of the sth replicating claim. Note that sincethe a_(j,s)'s are non-negative (equation 11.2.3K), and the x_(j)'s arenon-negative (fills are always non-negative), y_(s) will also benon-negative.

To keep auction sponsor risk low, the DBAR embodiment may require thatthe total premium collected is exactly sufficient to payout thein-the-money filled orders, or the self-hedging condition. Note that, inequilibrium, the sth replicating claim has a filled amount of

$\begin{matrix}{y_{s} + \frac{\theta_{s}}{p_{s}}} & {11.4{.5}C}\end{matrix}$

for s=1, 2, . . . , 2S−2. Therefore, the self-hedging condition can bemathematically stated as

$\begin{matrix}{{\sum\limits_{s = 1}^{{2S} - 2}{{d^{s}(U)}\left( {y_{s} + \frac{\theta_{s}}{p_{s}}} \right)}} = M} & {11.4{.5}D}\end{matrix}$

for all values of U.

As described in Appendix 11B, the self-hedging condition is equivalentto

$\begin{matrix}{{{y_{s} + \frac{\theta_{s}}{p_{s}}} = M}{s = 1},2,\ldots \mspace{14mu},{{2S} - 2}} & {11.4{.5}E}\end{matrix}$

Note that the auction sponsor takes on risk to the underlying onlythrough P&L in the opening orders.

Equation 11.4.5E relates y_(s), the aggregated filled amount of the sthreplicating claim, and p_(s), the price of the sth replicating claim.For M and θ_(s) fixed, the greater y_(s), then the higher p_(s) and thehigher the prices of strategies that pay out if the sth replicatingclaim expires in-the-money. Similarly, the lower the customer payoutsy_(s), then the lower p_(s) and the lower the prices of derivatives thatpay out if the sth replicating claim expires in-the-money. Thus, in thispricing framework, the demand by customers for a particular replicatingclaim is closely related to the price for that replicating claim.

Let m_(s) denote the total filled premium associated with replicatingclaim s. Then

m _(s) ≡p _(s) y _(s)θ_(s) +s=1, 2, . . . , 2S−2  11.4.5F

Substituting this definition into Equation 11.4.5E gives that

m _(s) =M _(p) _(s) s=1, 2, . . . , 2S−2  11.4.5G

For the with replication claim, one can also write

m _(v) =M _(p) _(v) v=1, 2, . . . , 2S−2  11.4.5H

Note that all quantities in equation 11.4.5H are strictly positive.Therefore, dividing 11.4.5G by 11.4.5H gives that

$\begin{matrix}{{\frac{m_{s}}{m_{v}} = \frac{p_{s}}{p_{v}}}s,{v = 1},2,\ldots \mspace{14mu},{{2S} - 2}} & {11.4{.5}I}\end{matrix}$

Thus, in the DBAR equilibrium, the relative amounts of premium investedin any two replicating claims equal the relative prices of thecorresponding replicating claims.

11.4.6 Maximizing Premium to Determine the DBAR Equilibrium

In determining the equilibrium, the auction sponsor may seek to maximizethe total filled premium M subject to the constraints described above.Combining all of the above equations to express the DBAR equilibriummathematically gives the following

$\begin{matrix}\begin{matrix}{{maximize}\mspace{14mu} M} & \; \\{{subject}\mspace{14mu} {to}} & \; \\{0 < p_{s}} & {{s = 1},2,\ldots \mspace{14mu},{{2S} - 2}} \\{{\sum\limits_{s = 1}^{{2S} - 2}p_{s}} = 1} & \; \\{\pi_{j} \equiv {\sum\limits_{s = 1}^{{2S} - 2}{a_{j,s}p_{s}}}} & {{j = 1},2,\ldots \mspace{14mu},J} \\\left. \begin{matrix}{\left. {w_{j}^{a} < \pi_{j}}\rightarrow x_{j} \right. = 0} \\{w_{j}^{a} = \left. \pi_{j}\rightarrow{0 \leq x_{j} \leq r_{j}} \right.} \\{\left. {w_{j}^{a} > \pi_{j}}\rightarrow x_{j} \right. = r_{j}}\end{matrix} \right\} & {{j = 1},2,\ldots \mspace{14mu},J} \\{y_{s} \equiv {\sum\limits_{j = 1}^{J}\; {a_{j,s}x_{j}}}} & {{s = 1},2,\ldots \mspace{14mu},{{2S} - 2}} \\{M \equiv {\left( {\sum\limits_{j = 1}^{J}\; {x_{j}\pi_{j}}} \right) + {\sum\limits_{s = 1}^{{2S} - 2}\; \theta_{s}}}} & \; \\{{y_{s} + \frac{\theta_{s}}{p_{s}}} = M} & {{s = 1},2,\ldots \mspace{14mu},{{2S} - 2}}\end{matrix} & {11.4{.6}A}\end{matrix}$

This maximization of M can be solved using mathematical programmingmethods of section 7.9, with changes in the number of replicating claims(2S−2) and changes in the formula for the adjusted limit price (section11.4.4).

11.5 Examples of DBAR Equilibria with the Digital Replicating Basis andthe Vanilla Replicating Basis

This section illustrates the techniques discussed above with twoexamples. Section 11.5.1 describes the underlying and the customerorders. Section 11.5.2 discusses the equilibrium using the digitalreplicating basis of section 11.1. Section 11.5.3 analyzes theequilibrium with the vanilla replicating basis of section 11.2.

For notation, let C^(θ)(U) denote the opening order P&L for the auctionsponsor. The opening order P&L C^(θ)(U) is written as a function of U toexplicitly express its dependence on the outcome of the underlying. Inan example embodiment, the auction sponsor may be exposed to bothopening order P&L C^(θ)(U) and replication P&L C^(R)(U) simultaneously.Let this combined quantity be called the outcome dependent P&L, denotedas C^(T)(U), and computed as follows

C ^(T)(U)=C ^(θ)(U)+C ^(R)(U)  11.5A

Once again, C^(T) takes on the argument U to show that it is outcomedependent.

11.5.1 Auction Set-Up

The auction set-up is as follows. The underlying U for the DBAR auctionis the equity price of a US company. Consistent with the recentdecimalization of the NYSE, the minimum change in the underlying equityprice is 0.01, denoted by p. The auction sponsor initially investsΘ=800. Customers trade derivatives strategies based on the followingthree strikes: k₁=40, k₂=50, and k₃=60. Customers pay premium for filledorders on the same date as the auction sponsor pays out in-the-moneyclaims. For this example, it is assumed that there is no finite lowerbound k₀ for this underlying.

Customers submit a total of J=3 orders in this DBAR auction, describedin table 11.5.1. Column two of table 11.5.1 shows the derivativesstrategy for each customer order. Note that there is one customer orderfor a digital strategy and two customer orders for vanilla strategies.Column three contains the payout function d_(j) for each derivativesstrategy, and column four shows the requested number of contracts r_(j)by each customer. For the digital order, one contract pays out $1 if theoption expires in-the-money. For both vanilla orders, assume that onecontract pays out $1 per 1 unit that the strategy expires in-the-money.Column five shows the customer's limit price w_(j) per contract. Thelimit price represents the maximum price the customer is willing to payper contract for the requested derivatives strategy.

TABLE 11.5.1 Details of the customer orders. Requested Number LimitPrice Strategy Derivatives Payout Per Contract of Contracts Per Contractj Strategy d_(j)(U) r_(j) w_(j) 1 50-60 Digital Range${d_{1}(U)} = \left\{ \begin{matrix}0 & {U < 50} \\1 & {50 \leq U < 60} \\0 & {50 \leq U}\end{matrix} \right.$ 1,000,000 0.3 2 50-40 Vanilla Put Spread${d_{2}(U)} = \left\{ \begin{matrix}10 & {U < 40} \\{50 - U} & {40 \leq U < 50} \\0 & {50 \leq U}\end{matrix} \right.$ 200,000 6 3 50-60 Vanilla Call Spread${d_{3}(U)} = \left\{ \begin{matrix}0 & {U < 50} \\{U - 50} & {50 \leq U < 60} \\10 & {60 \leq U}\end{matrix} \right.$ 200,000 4

11.5.2 The DBAR Equilibrium Based on the Digital Replicating Basis

This section analyzes the DBAR equilibrium with three customer ordersusing the digital replicating basis discussed in section 11.1.

Table 11.5.2A shows the opening orders for this DBAR auction. Asdescribed above, the auction sponsor initially invests $800. Asdisplayed in the fourth column, $200 of opening orders is investedequally in each of the four digital replicating claims.

TABLE 11.5.2A Opening orders for the digital replicating basis. DigitalOpening Order Replicating Outcome Digital Premium Amount Claim s RangeReplicating Claim θ_(s) 1 U < 40 Digital put struck at 40 $200 2 40 ≦ U< 50 Digital range with $200 strikes of 40 and 50 3 50 ≦ U < 60 Digitalrange with $200 strikes of 50 and 60 4 60 ≦ U Digital call struck at 60$200

The auction sponsor can calculate the replication weights for thedigital range, the vanilla put spread, and the vanilla call spread usingequations 11.1.3D, 11.1.3H, and 11.1.3F, respectively. These replicationweights are displayed in table 11.5.2B.

TABLE 11.5.2B Replication weights for the customer orders using thedigital replicating basis. Strategy j Derivatives Strategy a_(j,1)a_(j,2) a_(j,3) a_(j,4) 1 50-60 Digital Range 0 0 1 0 2 50-40 VanillaPut Spread 10 5.005 0 0 3 50-60 Vanilla Call Spread 0 0 4.995 10

Based on the three customer orders, the opening orders, and thereplicating weights, the auction sponsor can solve for the DBARequilibrium. Table 11.5.2C shows the DBAR equilibrium prices, fills, andpremiums paid for the customer orders and table 11.5.2D displaysequilibrium information for the opening orders.

TABLE 11.5.2C Equilibrium information for the customer orders using thedigital replicating basis. Equilibrium Equilibrium EquilibriumDerivatives Price Fill Amount Premium Paid Strategy j Strategy π_(j)x_(j) x_(j) π_(j) 1 50-60 Digital 0.134401 1,000,000 $134,401 Range 250-40 Vanilla 5.326330 200,000 $1,065,266 Put Spread 3 50-60 Vanilla4.000000 199,978 $799,910 Call Spread

TABLE 11.5.2D Equilibrium information for the opening orders using thedigital replicating basis. Digital Equilibrium Price EquilibriumReplicating Per $1 USD Payout Fill Amount Claim s p_(s) θ_(s)/p_(s) 10.532532449 376 2 0.000200125 999,376 3 0.134400500 1,488 4 0.332866926601Based on this equilibrium information, the auction sponsor can computethe opening order P&L C^(θ)(U). Since the payout for the sth digitalreplication claim is θ_(s)/p_(s) if that claim expires in-the-money,C^(θ)(U) can be computed as follows for s=1, 2, . . . , S

$\begin{matrix}{{{C^{\theta}(U)} = {\frac{\theta_{s}}{p_{s}} - \Theta}}{k_{s - 1} \leq U < k_{s}}} & {11.5{.2}A}\end{matrix}$

To illustrate, assume that the outcome of the underlying is 59, whichoccurs in state s=3. The opening order P&L if outcome 59 occurs isdenoted as C^(θ)(59). Using equation 11.5.2A, one can determine that

$\begin{matrix}\begin{matrix}{{C^{\theta}(59)} = {\frac{\theta_{3}}{p_{3}} - \Theta}} \\{= {\frac{\$ 200}{0.134400500} - {\$ 800}}} \\{= {\$ 688}}\end{matrix} & {11.5{.2}B}\end{matrix}$

The replication P&L C^(R)(59) can be computed based on equation 11.1.4Bas

$\begin{matrix}\begin{matrix}{{C^{R}(59)} = {\sum\limits_{s = 1}^{S}{\sum\limits_{j = 1}^{J}{{{I\left\lbrack {k_{s - 1} \leq 59 < k_{s}} \right\rbrack}\left\lbrack {a_{j,s} - {d_{j}(59)}} \right\rbrack}x_{j}}}}} \\{= {\sum\limits_{j = 1}^{J}{\left\lbrack {a_{j,3} - {d_{j}(59)}} \right\rbrack x_{j}}}} \\{= {\left\lbrack {a_{3,3} - {d_{3}(59)}} \right\rbrack x_{3}}} \\{= {\left( {{{\$ 4}{.995}} - {\$ 9}} \right)\left( {199,978} \right)}} \\{= \left( {{\$ 800},912} \right)}\end{matrix} & {11.5{.2}C}\end{matrix}$

Therefore, the total outcome dependent P&L is

$\begin{matrix}\begin{matrix}{{C^{T}(59)} = {{C^{\theta}(59)} + {C^{R}(59)}}} \\{= {{\$ 688} - {{\$ 800},912}}} \\{= \left( {{\$ 800},224} \right)}\end{matrix} & {11.5{.2}D}\end{matrix}$

Based on this approach, the auction sponsor can compute summarystatistics for the opening order P&L, the replication P&L, and theoutcome dependent P&L for this auction equilibrium.

Table 11.5.2E shows several such summary statistics. Row one and row twodisplay the minimum and maximum values, respectively, while theremaining rows show various probability weighted measures. To computethe probability of a specific outcome with the digital replicatingbasis, the auction sponsor might apply the intrastate uniform model,which assumes the probability of state s occurring equals p_(s) andevery outcome is equally likely to occur within a state. In this case,the probability of a specific outcome is

$\begin{matrix}{{\Pr \left\lbrack {U = u} \right\rbrack} = {{\frac{p_{s}\rho}{k_{s} - k_{s - 1}}\mspace{14mu} k_{s - 1}} \leq u < k_{s}}} & {11.5{.2}E}\end{matrix}$

These probability weighted computations are discussed in Appendix 11C.Note that the minimum outcome dependent P&L is ($998,199), and furthernote that the probability that the outcome dependent P&L is less thanzero is 93.26%.

Examining table 11.5.2E, it is worth observing the following. Althoughoutcome dependent P&L is the sum of opening order P&L and replicationP&L (equation 11.5A), the summary statistic for outcome dependent P&L(in column four of table 11.5.2E) will not necessarily be the sum of thecorresponding summary statistic for opening order P&L (column two oftable 11.5.2E) and replication P&L (column three of table 11.5.2E). Forexample, the standard deviation of the outcome dependent P&L ($212,265)does not equal the standard deviation of the opening order P&L ($14,133)plus the standard deviation of the replication P&L ($211,794). This istrue for many of the summary statistics in table 11.5.2E and table11.5.3E.

TABLE 11.5.2E Summary statistics for opening order P&L C^(θ)(U),replication P&L C^(R)(U), and outcome dependent P&L C^(T)(U) for DBARauction using the digital replicating basis. Opening Order ReplicationTotal Outcome Summary P&L P&L Dependent P&L Statistic C^(θ)(U) C^(R)(U)C^(T)(U) Minimum ($424) ($999,000) ($998,199) Maximum $998,576   $999,000 $1,997,576   Probability < 0 86.54% 6.73% 93.26%  Probability =0  0.00% 86.54%  0.00% Probability > 0 13.46% 6.73% 6.74% Average  $0    $0     $0 Standard Deviation $14,133   $211,794 $212,265Semi-Standard $381 $299,522 $160,300 Deviation Skewness    70.6      0.0       0.1

11.5.3 The DBAR Equilibrium Based on the Vanilla Replicating Basis

This section examines the equilibrium based on the vanilla replicatingbasis discussed in section 11.2. Table 11.5.3A shows the opening ordersfor these replicating claims. As before, the auction sponsor allocates atotal of Θ=800 in opening orders.

TABLE 11.5.3A Opening orders for the vanilla replicating basis. VanillaOpening Order Replicating Outcome Vanilla Premium Claim s RangeReplicating Claim Amount θ_(s) 1 U < 40 Digital put struck at 40 $200 240 ≦ U < 50 Rescaled vanilla put struck $100 at 50 knockout at 39.99 340 ≦ U < 50 Rescaled vanilla call struck $100 at 40 knockout at 50 4 50≦ U < 60 Rescaled vanilla put struck $100 at 60 knockout at 49.99 5 50 ≦U < 60 Rescaled vanilla call struck $100 at 50 knockout at 60 6 60 ≦ UDigital call struck at 60 $200

The auction sponsor can calculate the replication weights for thedigital range, the vanilla put spread, and the vanilla call spread usingequations 11.2.4I, 11.2.5K, and 11.2.5H, respectively. These replicationweights are displayed in table 11.5.3B.

TABLE 11.5.3B Replicating weights for the customer orders using thevanilla replicating basis. Strategy j Derivatives Strategy a_(j,1)a_(j,2) a_(j,3) a_(j,4) a_(j,5) a_(j,6) 1 50-60 Digital Range 0 0 0 1 10 2 50-40 Vanilla Put Spread 10 10 0 0 0 0 3 50-60 Vanilla Call Spread 00 0 0 10 10

Based on the customer orders, the opening orders, and the replicatingweights, the auction sponsor can solve for the DBAR equilibrium. Table11.5.3C displays the DBAR equilibrium prices, fills, and premiums paidfor the customer orders and table 11.5.3D shows equilibrium informationfor the opening orders.

In comparing tables 11.5.3C and 11.5.3D with tables 11.5.2C and 11.5.2D,respectively, note that the equilibrium prices and fills using thevanilla replicating basis are different than those from the digitalreplicating basis even though the customer orders are identical for bothcases.

TABLE 11.5.3C Equilibrium information for the customer orders using thevanilla replicating basis. Equilibrium Equilibrium EquilibriumDerivatives Price Fill Amount Premium Paid Strategy j Strategy π_(j)x_(j) x_(j) π_(j) 1 50-60 Digital 0.30 1,666 $500 Range 2 50-40 Vanilla5.999 200,000 $1,199,800 Put Spread 3 50-60 Vanilla 4.00 199,850 $799,400 Call Spread

TABLE 11.5.3D Equilibrium information for the opening orders using thevanilla replicating basis. Vanilla Equilibrium Price EquilibriumReplicating Per $1 USD Payout Fill Amount Claim s p_(s) θ_(s)/p_(s) 10.399933460 500 2 0.199966730 500 3 0.000049988 2,000,500 4 0.0000500291,998,834 5 0.299950032 333 6 0.100049761 1,999

As before, let C^(θ)(U) denote the opening order P&L for the auctionsponsor. For the vanilla replicating basis, C^(θ)(U) can be computed as

$\begin{matrix}\begin{matrix}{{C^{\theta}(U)} = {\left( {\sum\limits_{s = 1}^{{2S} - 2}{{d^{s}(U)}\frac{\theta_{s}}{p_{s}}}} \right) - \Theta}} \\{= {\frac{{I\left\lbrack {U < k_{1}} \right\rbrack}\theta_{1}}{p_{1}} +}} \\{{{\sum\limits_{s = 2}^{S - 1}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left( {{{d^{{2s} - 2}(U)}\frac{\theta_{{2s} - 2}}{p_{{2s} - 2}}} + {{d^{{2s} - 1}(U)}\frac{\theta_{{2s} - 1}}{p_{{2s} - 1}}}} \right)}} +}} \\{{{{I\left\lbrack {U \geq k_{S - 1}} \right\rbrack}\frac{\theta_{{2S} - 2}}{p_{{2S} - 2}}} - \Theta}}\end{matrix} & {11.5{.3}A}\end{matrix}$

C^(θ)(U) is the difference between the opening order payouts and thetotal invested amount in the opening orders.

To illustrate, assume again that the outcome of the underlying is 59.Therefore,

$\begin{matrix}\begin{matrix}{{C^{\theta}(59)} = {\left( {\sum\limits_{s = 1}^{{2S} - 2}{{d^{s}(59)}\frac{\theta_{s}}{p_{s}}}} \right) - \Theta}} \\{= {\frac{{d^{4}(59)}\theta_{4}}{p_{4}} + \frac{{d^{5}(59)}\theta_{5}}{p_{5}} - {\$ 800}}} \\{= {\frac{(0.1) \times ({\$ 100})}{0.000050029} + \frac{(0.9) \times ({\$ 100})}{0.299950032} - {\$ 800}}} \\{= {{{\$ 199},884} + {\$ 300} - {\$ 800}}} \\{= {{\$ 199},384}}\end{matrix} & {11.5{.3}B}\end{matrix}$

As shown by the general replication theorem in section 11.2.3, thereplication P&L C^(R)(59) equals zero. Therefore, by 11.5A, the outcomedependent P&L C^(T)(59) is

$\begin{matrix}\begin{matrix}{{C^{T}(59)} = {{C^{\theta}(59)} + {C^{R}(59)}}} \\{= {{\$ 199},384}}\end{matrix} & {11.5{.3}C}\end{matrix}$

Similar to table 11.5.2E, table 11.5.3E shows several summary statisticsfor the opening order P&L, the replication P&L, and the outcomedependent P&L. Here, the probability weighted statistics are computedusing the assumption that

$\begin{matrix}{{\Pr \left\lbrack {U = u} \right\rbrack} = {{\frac{\left( {p_{{2s} - 2} + p_{{2s} - 1}} \right)\rho}{k_{s} - k_{s - 1}}\mspace{14mu} k_{s - 1}} \leq u < k_{s}}} & {11.5{.3}D}\end{matrix}$

for s=2, 3, . . . , S−1. This is discussed in further detail in Appendix11C. Note that the minimum outcome dependent P&L is ($300), whichcompares favorably to ($998,199), the minimum outcome dependent P&Lusing the digital replicating basis. Further note that the probabilitythat outcome dependent P&L is less than zero has dropped to 40.01% from93.26% using the digital replicating basis.

TABLE 11.5.3E Summary statistics for opening order P&L C^(θ)(U),replication P&L C^(R)(U), and outcome dependent P&L C^(T)(U) using thevanilla replicating basis. Opening Order Replication Total OutcomeSummary P&L P&L Dependent Statistic C^(θ)(U) C^(R)(U) P&L C^(T)(U)Minimum    ($300) 0    ($300) Maximum $1,998,034   0 $1,998,034  Probability < 0 40.01% 0 40.01% Probability = 0  0.00% 0  0.00%Probability > 0 59.99% 0 59.99% Average $499,692 0 $499,692 StandardDeviation $625,568 0 $625,568 Semi-Standard Deviation $531,623 0$531,623 Skewness       0.5 0       0.5

11.6 Replication Using the Augmented Vanilla Replicating Basis

As shown in equation 11.2.31, the replication P&L for a buy of strategyd is

C ^(R)(U)=β₁(U−E[U|U<k ₁])I[U<k ₁]+β_(S)(E[U|U≧k _(S−1) ]−U)I[U≧k_(S−1)]  11.6A

If β₁ or β_(S) is non-zero, then the auction sponsor's replication P&Lmay be unbounded. For example, a vanilla call has β₁=0 and β_(S)=1, andtherefore

C ^(R)(U)=(E[U|U≧k _(S−1) ]−U)I[U≧k _(S−1)]  11.6B

If U is very large, then the auction sponsor can lose an unboundedamount of money on a customer purchase of a vanilla call. To eliminatereplication P&L in this case, this section describes the augmentedvanilla replicating basis.

Section 11.6.1 introduces the augmented vanilla replicating basis.Section 11.6.2 describes the general replication theorem using thisaugmented basis. Section 11.6.3 uses this theorem to compute replicatingweights for digital and vanilla options, and section 11.6.4 discussesthe mathematical restrictions based on this equilibrium.

11.6.1 The Augmented Vanilla Replicating Basis

The augmented vanilla replicating basis includes the 2S−2 replicatingclaims from the vanilla replicating basis of section 11.2.2 plus twoadditional replicating claims. For notation, the additional replicatingclaims will be the 1^(st) replicating claim and the 2Sth replicatingclaim.

The 1^(st) augmented vanilla replicating claim is the vanilla put struckat k₁−ρ. This claim has the payout function

$\begin{matrix}{{d^{1}(U)} = \left\{ \begin{matrix}{k_{1} - \rho - U} & {U < {k_{1} - \rho}} \\0 & {{k_{1} - \rho} \leq U}\end{matrix} \right.} & {11.6{.1}A}\end{matrix}$

The 2^(nd), 3^(rd), . . . , 2S−1^(st) augmented vanilla replicatingclaims are identical to the 1^(st), 2^(nd), 2S−2^(nd) vanillareplicating claims, respectively, from section 11.2.2. The 2Sthaugmented vanilla replicating claim is the vanilla call struck atk_(S−1), which has a payout of

$\begin{matrix}{{d^{2S}(U)} = \left\{ \begin{matrix}0 & {U < k_{S - 1}} \\{U - k_{S - 1}} & {k_{S - 1} \leq U}\end{matrix} \right.} & {11.6{.1}B}\end{matrix}$

Table 11.6.1 shows the 2S claims for the augmented vanilla replicatingbasis.

TABLE 11.6.1 The payout ranges and replicating claims for the augmentedvanilla replicating basis. Claim Payout Augmented Vanilla EuropeanNumber Range Replicating Claim Knockout? 1 U < k₁ − ρ Vanilla put struckat k₁ − ρ None 2 U < k₁ Digital put struck at k₁ None 3 k₁ ≦ U < k₂Rescaled vanilla put struck at k₂ Knockout at k₁ − ρ 4 k₁ ≦ U < k₂Rescaled vanilla call struck at k₁ Knockout at k₂ 5 k₂ ≦ U < k₃ Rescaledvanilla put struck at k₃ Knockout at k₂ − ρ 6 k₂ ≦ U < k₃ Rescaledvanilla call struck at k₂ Knockout at k₃ . . . . . . . . . . . . 2s − 1k_(s−1) ≦ U < k_(s) Rescaled vanilla put struck at k_(s) Knockout atk_(s−1) − ρ 2s k_(s−1) ≦ U < k_(s) Rescaled vanilla call struck atk_(s−1) Knockout at k_(s) 2s + 1 k_(s) ≦ U < k_(s+1) Rescaled vanillaput struck at k_(s+1) Knockout at k_(s) − ρ 2s + 2 k_(s) ≦ U < k_(s+1)Rescaled vanilla call struck at k_(s) Knockout at k_(s+1) . . . . . . .. . . . . 2S − 3 k_(S−2) ≦ U < k_(S−1) Rescaled vanilla put struck atk_(S−1) Knockout at k_(S−2) − ρ 2S − 2 k_(S−2) ≦ U < k_(S−1) Rescaledvanilla call struck at k_(S−2) Knockout at k_(S−1) 2S − 1 k_(S−1) ≦ UDigital call struck at k_(S−1) None 2S k_(S−1) ≦ U Vanilla call struckat k_(S−1) None

11.6.2 The General Replicating Theorem for the Augmented VanillaReplicating Basis

For notation, let d″ and d″ denote functions of the derivatives strategyd computed as follows

$\begin{matrix}{{\overset{\_}{d}}^{''} = {\max\limits_{{({k_{1} - \rho})} \leq U \leq k_{S - 1}}{d(U)}}} & {11.6{.2}A} \\{{\underset{\_}{d}}^{''} = {\min\limits_{{({k_{1} - \rho})} \leq U \leq k_{S - 1}}{d(U)}}} & {11.6{.2}B}\end{matrix}$

The following theorem shows how to construct the replicating weights(a₁, a₂, . . . , a_(2S−1), a_(2S)) of strategy d.

General Replication Theorem. Under assumptions 1, 3, 4, and 5 of section11.2.1, the replicating weights for a buy of strategy d are

a ₁=−β₁  11.6.2C

a ₂=α₁+β₁(k ₁−ρ)− d″  11.6.2D

a _(2s−1)=α_(s)+β_(s) k _(s−1) −d″ s=2, 3, . . . , S−1  11.6.2E

a _(2s)=α_(s)+β_(s) k _(s) −d″ s=2, 3, . . . , S−1  11.6.2F

a _(2s−1)=α_(S) −d″  11.6.2G

a_(2S)=β_(S)  11.6.2H

For a sell of strategy d, the replicating weights are

a₁=β₁  11.6.2I

a ₂ = d″−α ₁−β₁(k ₁−ρ)  11.6.2J

a _(2s−1) = d″−α _(s)−β_(s) k _(s−1) s=2, 3, . . . , S−1  11.6.2K

a _(2s) = d″−α _(s)−β_(s) k _(s) s=2, 3, . . . , S−1  11.6.2L

a _(2S−1) = d″−α _(s)  11.6.2M

a _(2S)=−β_(S)  11.6.2N

The proof of this theorem follows closely the proof in appendix 11A.Note that this theorem does not require assumption 2 to hold.

Using the augmented vanilla replicating basis, the replication P&LC^(R)(U) for a buy or sell of d is 0, regardless of the outcome of U.

Note that for any s=2, 3, . . . , 2S−1, the replicating weight for a buyof d plus the replicating weight for a sell of d equals d″−d″. Note thatfor s=1 or s=2S, the replicating weight for a buy of d plus thereplicating weight for a sell of d equals zero.

The replication weights for a buy of d or a sell of d satisfy

min(a ₂ , a ₃ , . . . , a _(2S−2) , a _(2S−1))=0  11.6.20

ensuring that y₂, y₃, . . . , y_(2s−2), y_(2S−1) (as defined in section11.6.4) are also non-negative. However, a₁ and a_(2S) can be negativeand section 11.6.4 shows how y₁ and y_(2S) are restricted to benon-negative.

11.6.3 Computing Replicating Weights for Digital and Vanilla Derivatives

Consider the special case where as defined in assumption 4, d satisfies

β₁=β_(s)=0  11.6.3A

In this case, the payout of the derivatives strategy is constant andequal to α₁ if the underlying U is less than k₁. Similarly, the payoutof the derivatives strategy is constant and equal to α_(S) if theunderlying U is greater than or equal to k_(S−1). Strategies thatsatisfy 11.6.3A include digital calls, digital puts, range binaries,vanilla call spreads, vanilla put spreads, collared vanilla straddles,and collared forwards. Applying the theorem above, then, equation11.6.3A implies that

a₁=a_(2S)=0  11.6.3B

For these instruments, the remaining replicating weights (a₂, a₃, . . ., a_(2S−2), a_(2S−1)) correspond to the replicating weights (a₁, a₂, . .. , a_(2S−3), a_(2S−2)) defined for these instruments in sections 11.2.4and 11.2.5.

To compute the replication weights for a vanilla call, recall that thepayout function d for a vanilla call with strike k_(v) is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}0 & {U < k_{v}} \\{U - k_{v}} & {k_{v} \leq U}\end{matrix} \right.} & {11.6{.3}C}\end{matrix}$

Note that d″=0 and d″=k_(S−1)−k_(v). For a buy order, the replicationweights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {{s = 1},2,\ldots \mspace{14mu},{2v}} \\{k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} - k_{v}} & {{s = {{2v} + 1}},{{2v} + 2},\ldots \mspace{14mu},{{2S} - 2}} \\{k_{S - 1} - k_{v}} & {s = {{2S} - 1}} \\1 & {s = {2S}}\end{matrix} \right.} & {11.6{.3}D}\end{matrix}$

For a sell order for a vanilla call with strike k_(v) the replicationweights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}0 & {s = 1} \\{k_{S - 1} - k_{v}} & {{s = 2},3,\ldots \mspace{14mu},{2v}} \\{k_{S - 1} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = {{2v} + 1}},{{2v} + 2},\ldots \mspace{14mu},{{2S} - 2}} \\0 & {s = {{2S} - 1}} \\{- 1} & {s = {2S}}\end{matrix} \right.} & {11.6{.3}E}\end{matrix}$

The payout function d for a vanilla put with strike k_(v) is

$\begin{matrix}{{d(U)} = \left\{ \begin{matrix}{k_{v} - U} & {U < k_{v}} \\0 & {k_{v} \leq U}\end{matrix} \right.} & {11.6{.3}F}\end{matrix}$

Note that in this case, d″=0 and d=k_(v)−k₁+ρ. For a buy order for avanilla put with strike k_(v) the replication weights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}{- 1} & {s = 1} \\{k_{v} - k_{1} + \rho} & {s = 2} \\{k_{v} - k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}}} & {{s = 3},4,\ldots \mspace{14mu},{2v}} \\0 & {{s = {{2v} + 1}},{{2v} + 2},\ldots \mspace{14mu},{2S}}\end{matrix} \right.} & {11.6{.3}G}\end{matrix}$

For a sell order for a vanilla put with strike k_(v) the replicationweights are

$\begin{matrix}{a_{s} = \left\{ \begin{matrix}1 & {s = 1} \\0 & {s = 2} \\{k_{{int}{\lbrack{{({s + 1})}/2}\rbrack}} - k_{1} + \rho} & {{s = 3},4,\ldots \mspace{14mu},{2v}} \\{k_{v} - k_{1} + \rho} & {{s = {{2v} + 1}},{{2v} + 2},\ldots \mspace{14mu},{{2S} - 1}} \\0 & {s = {2S}}\end{matrix} \right.} & {11.6{.3}H}\end{matrix}$

The auction sponsor can compute the replication weights for straddlesand forwards using the above theorem in a similar manner.

11.6.4 Mathematical Restrictions for the Equilibrium Based on theAugmented Vanilla Replicating Basis

This section discusses the mathematical restrictions for pricing andfilling customer orders in such a DBAR equilibrium. This section followsclosely the discussion in section 11.4.

Opening Orders and Customer Orders

The auction sponsor may enter opening orders θ_(s) for each of the 2Saugmented vanilla replicating claims whereby

θ_(s)>0 s=1, 2, . . . , 2S  11.6.4A

Assume that customers submit a total of J orders in the auction, indexedby j=1, 2, . . . , J. For customer order j, let r_(j) denote therequested number of contracts, let w_(j) denote the limit price, and letd_(j) denote the payout function.

Pricing and Filling Derivatives Strategies

Let p_(s) denote the equilibrium price of the sth augmented vanillareplicating claim s=1, 2, . . . , 2S. Mathematically, the auctionsponsor may require that

$\begin{matrix}{{{p_{s} > {0\mspace{14mu} s}} = 1},2,\ldots \mspace{14mu},{2S}} & {11.6{.4}B} \\{{\sum\limits_{s = 2}^{{2S} - 1}p_{s}} = 1} & {11.6{.4}C}\end{matrix}$

Here, the auction sponsor requires that the prices of the augmentedvanilla replicating claims are positive and sum to one. Note that p₁ andp_(2S) are not part of the summation in equation 11.6.4C.

Based on these prices, the auction sponsor may determine the equilibriumprice π_(j) of each derivatives strategy as

$\begin{matrix}{\pi_{j} \equiv {\sum\limits_{s = 1}^{2S}{a_{j,s}p_{s}}}} & {11.6{.4}D}\end{matrix}$

where a_(j,s) is the replicating weight for customer order j foraugmented replicating claim s, computed based on the theorem in section11.6.2.

Define d_(j) ″ and d_(j) ″ as the analogues to d″ and d″ for customerorder j, respectively. Let w_(j) ^(a) denote the adjusted limit pricefor customer order j. If order j is a buy of strategy d_(j), then

w _(j) ^(a) =w _(j)− d _(j) ″  11.6.4E

If order j is a sell of strategy d_(j), then

w _(j) ^(a)= d _(j) ″−w _(j)  11.6.4F

The auction sponsor can employ the logic of equation 11.4.4C to fillcustomer orders.

The DBAR Equilibrium Conditions

Let M denote the total replicated premium paid in the auction

$\begin{matrix}{M \equiv {\left( {\sum\limits_{j = 1}^{J}{x_{j}\pi_{j}}} \right) + {\sum\limits_{s = 1}^{2S}\theta_{s}}}} & {11.6{.4}G}\end{matrix}$

Define y_(s) as

$\begin{matrix}{y_{s} \equiv {\sum\limits_{j = 1}^{J}{a_{j,s}x_{j}}}} & {11.6{.4}H}\end{matrix}$

for s=1, 2, . . . , 2S. To keep risk low, the auction sponsor mayenforce the condition

$\begin{matrix}{{{y_{s} + \frac{\theta_{s}}{p_{s}}} = {{M\mspace{56mu} s} = 2}},3,\ldots \mspace{14mu},{{2S} - 1}} & {11.6{.4}I}\end{matrix}$

In addition, to eliminate risk for the 1^(st) and 2Sth replicating claimthe auction sponsor may require that

y₁=y_(2S)=0  11.6.4J

In determining the equilibrium, the auction sponsor may seek to maximizethe total filled premium M subject to the constraints described above.Combining all of these equations to express the DBAR equilibriummathematically gives the following

$\begin{matrix}{{{{\left. {{{maximize}\mspace{14mu} M}{{subject}\mspace{14mu} {to}}{{{0 < {p_{s}\mspace{225mu} s}} = 1},2,\ldots \mspace{14mu},{2S}}{{\sum\limits_{s = 2}^{{2S} - 1}p_{s}} = 1}{{{\pi_{j} \equiv {\sum\limits_{s = 1}^{2S}{a_{j,s}p_{s}\mspace{135mu} j}}} = 1},2,\ldots \mspace{14mu},J}\begin{matrix}{\left. {w_{j}^{a} < \pi_{j}}\rightarrow x_{j} \right. = 0} \\{w_{j}^{a} = \left. \pi_{j}\rightarrow{0 \leq x_{j} \leq r_{j}} \right.} \\{\left. {w_{j}^{a} > \pi_{j}}\rightarrow x_{j} \right. = r_{j}}\end{matrix}} \right\} \mspace{34mu} j} = 1},2,\ldots \mspace{14mu},J}{{{y_{s} \equiv {\sum\limits_{j = 1}^{J}{a_{j,s}x_{j}\mspace{135mu} s}}} = 1},2,\ldots \mspace{14mu},{2S}}{M \equiv {\left( {\sum\limits_{j = 1}^{J}{x_{j}\pi_{j}}} \right) + {\sum\limits_{s = 1}^{2S}\theta_{s}}}}{{{y_{s} + \frac{\theta_{s}}{p_{s}}} = {{M\mspace{146mu} s} = 2}},3,\ldots \mspace{14mu},{{2S} - 1}}{y_{1} = {y_{2S} = 0}}} & {11.6{.4}K}\end{matrix}$

This maximization of M can be solved using mathematical programmingmethods based on section 7.9.

Appendix 11A: Proof of General Replication Theorem in Section 11.2.3

The proof of this theorem proceeds in two parts. Section 11A.1 derivesthe result for a buy of strategy d and then section 11A.2 derives theresult for a sell of strategy d. Similar to section 11.1.4, let e(U)denote the payout on the replicating portfolio based on weights (a₁, a₂,. . . , a_(2S-3), a_(2S-2))

$\begin{matrix}{{e(U)} = {\sum\limits_{s = 1}^{{2S} - 2}{a_{s}{d^{s}(U)}}}} & {11A{.1}}\end{matrix}$

The replication P&L C^(R)(U) versus d(U) minus d is

C ^(R)(U)=e(U)−(d(U)− d )  11A.2

Here d is subtracted from the payout function d to make the replicationweights as small as possible while remaining non-negative (non-negativereplication weights are a requirement to construct the DBARequilibrium). These formulas will be used in the proof below.

11A.1: Proof for a Buy of the Derivatives Strategy d

The derivation below shows that the payout on the replicating portfolioe(U) is the variance minimizing portfolio for the derivatives strategyd(U) minus d for every outcome U. To prove this theorem, U's range isdivided into three mutually exclusive, collectively exhaustive cases.

Case 1: U<k₁. Note that the only replicating claim that pays out overthis range is the first replicating claim. In this case,

$\begin{matrix}\begin{matrix}{{e(U)} = {\sum\limits_{s = 1}^{{2S} - 2}{a_{s}{d^{s}(U)}}}} \\{= {a_{1}{d^{1}(U)}}} \\{= {\alpha_{1} + {\beta_{1}{E\left\lbrack U \middle| {U < k_{1}} \right\rbrack}} - \underset{\_}{d}}}\end{matrix} & {11A{.1}{.1}}\end{matrix}$

where the last step follows from the definition of a₁ in equation11.2.3A and the definition of d¹ in equation 11.2.2A. The payout on thederivatives strategy d over this range is

$\begin{matrix}\begin{matrix}{{d(U)} = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left( {\alpha_{s} + {\beta_{s}U}} \right)}}} \\{= {\alpha_{1} + {\beta_{1}U}}}\end{matrix} & {11A{.1}{.2}}\end{matrix}$

Therefore, the replication P&L versus d(U) minus d is

$\begin{matrix}\begin{matrix}{{C^{R}(U)} = {{e(U)} - \left( {{d(U)} - \underset{\_}{d}} \right)}} \\{= {\left( {\alpha_{1} + {\beta_{1}{E\left\lbrack U \middle| {U < k_{1}} \right\rbrack}} - \underset{\_}{d}} \right) - \left( {\alpha_{1} + {\beta_{1}U} - \underset{\_}{d}} \right)}} \\{= {\beta_{1}\left( {{E\left\lbrack U \middle| {U < k_{1}} \right\rbrack} - U} \right)}}\end{matrix} & {1A{.1}{.3}}\end{matrix}$

Since the estimate that minimizes the variance is that random variable'sexpected value, the replicating portfolio is variance minimizing.

Case 2:1c_(s−1)≦U<k_(s) for s=2, 3, . . . , S−1. When U is in thisrange, the only vanilla replicating claims that payout are the 2s−2^(nd)claim and the 2s−1^(st) claim. Therefore, the vanilla replicatingportfolio pays out

$\begin{matrix}\begin{matrix}{{e(U)} = {\sum\limits_{s = 1}^{{2S} - 2}{a_{s}{d^{s}(U)}}}} \\{= {{a_{{2s} - 2}{d^{{2s} - 2}(U)}} + {a_{{2s} - 1}{d^{{2s} - 1}(U)}}}} \\{= {{a_{{2s} - 2}\left( \frac{k_{s} - U}{k_{s} - k_{s - 1}} \right)} + {a_{{2s} - 1}\left( \frac{U - k_{s - 1}}{k_{s} - k_{s - 1}} \right)}}}\end{matrix} & {11A{.1}{.4}}\end{matrix}$

where the last step follows from the definitions of d^(2s−2) andd^(2s−1) from equations 11.2.2D and 11.2.2E. Substituting the values ofa_(2s−2) and a_(2s−1) from equations 11.2.3B and 11.2.3C into equation11A.1.4 gives

$\begin{matrix}\begin{matrix}{{e(U)} = {{a_{{2s} - 2}\left( \frac{k_{s} - U}{k_{s} - k_{s - 1}} \right)} + {a_{{2s} - 1}\left( \frac{U - k_{s - 1}}{k_{s} - k_{s - 1}} \right)}}} \\{= {{\left( {\alpha_{s} + {\beta_{s}k_{s - 1}} - \underset{\_}{d}} \right)\left( \frac{k_{s} - U}{k_{s} - k_{s - 1}} \right)} +}} \\{{\left( {\alpha_{s} + {\beta_{s}k_{s}} - \underset{\_}{d}} \right)\left( \frac{U - k_{s - 1}}{k_{s} - k_{s - 1}} \right)}}\end{matrix} & {11A{.1}{.5}}\end{matrix}$

Simplifying equation 11A.1.5 leads to

$\begin{matrix}\begin{matrix}{{e(U)} = {{\left( {\alpha_{s} + {\beta_{s}k_{s - 1}} - \underset{\_}{d}} \right)\left( \frac{k_{s} - U}{k_{s} - k_{s - 1}} \right)} +}} \\{{\left( {\alpha_{s} + {\beta_{s}k_{s}} - \underset{\_}{d}} \right)\left( \frac{U - k_{s - 1}}{k_{s} - k_{s - 1}} \right)}} \\{= {{\alpha_{s}\left( \frac{k_{s} - k_{s - 1}}{k_{s} - k_{s - 1}} \right)} + \left( \frac{{\beta_{s}k_{s - 1}k_{s}} - {\beta_{s}k_{s - 1}U}}{k_{s} - k_{s - 1}} \right) +}} \\{{\left( \frac{{\beta_{s}k_{s}U} - {\beta_{s}k_{s - 1}k_{s}}}{k_{s} - k_{s - 1}} \right) - {\underset{\_}{d}\left( \frac{k_{s} - k_{s - 1}}{k_{s} - k_{s - 1}} \right)}}} \\{= {\alpha_{s} + \frac{{\beta_{s}k_{s - 1}k_{s}} - {\beta_{s}k_{s - 1}U} + {\beta_{s}k_{s}U} - {\beta_{s}k_{s - 1}k_{s}}}{k_{s} - k_{s - 1}} - \underset{\_}{d}}} \\{= {\alpha_{s} + \frac{{\beta_{s}k_{s}U} - {\beta_{s}k_{s - 1}U}}{k_{s} - k_{s - 1}} - \underset{\_}{d}}} \\{= {\alpha_{s} + \frac{\beta_{s}{U\left( {k_{s} - k_{s - 1}} \right)}}{k_{s} - k_{s - 1}} - \underset{\_}{d}}} \\{= {\alpha_{s} + {\beta_{s}U} - \underset{\_}{d}}}\end{matrix} & {11A{.1}{.6}}\end{matrix}$

For this case, the payout on the derivatives strategy d over this rangeis

$\begin{matrix}\begin{matrix}{{d(U)} = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left( {\alpha_{s} + {\beta_{s}U}} \right)}}} \\{= {\alpha_{s} + {\beta_{s}U}}}\end{matrix} & {11A{.1}{.7}}\end{matrix}$

Therefore, the replication P&L versus d(U) minus d is

$\begin{matrix}\begin{matrix}{{C^{R}(U)} = {{e(U)} - \left( {{d(U)} - \underset{\_}{d}} \right)}} \\{= {\left( {\alpha_{s} + {\beta_{s}U} - \underset{\_}{d}} \right) - \left( {\alpha_{s} + {\beta_{s}U} - \underset{\_}{d}} \right)}} \\{= 0}\end{matrix} & {11A{.1}{.8}}\end{matrix}$

Thus the theorem holds over this range and the replication P&L C^(R)(U)is zero.

Case 3: U≧k_(S−1). Over this range, only the 2S−2^(nd) replicating claimpays out. Therefore,

$\begin{matrix}\begin{matrix}{{e(U)} = {\sum\limits_{s = 1}^{{2S} - 2}{a_{s}{d^{s}(U)}}}} \\{= {a_{{2S} - 2}{d^{{2S} - 2}(U)}}} \\{= {\alpha_{s} + {\beta_{s}{E\left\lbrack U \middle| {U \geq k_{s - 1}} \right\rbrack}\underset{\_}{d}}}}\end{matrix} & {11A{.1}{.9}}\end{matrix}$

where the last step follows from the definition of d^(2S−2) fromequation 11.2.2H. The payout on the derivatives strategy d over thisrange is

$\begin{matrix}\begin{matrix}{{d(U)} = {\sum\limits_{s = 1}^{S}{{I\left\lbrack {k_{s - 1} \leq U < k_{s}} \right\rbrack}\left( {\alpha_{s} + {\beta_{s}U}} \right)}}} \\{= {\alpha_{S} + {\beta_{S}U}}}\end{matrix} & {11A{.1}{.10}}\end{matrix}$

Therefore, the replication P&L is

$\begin{matrix}\begin{matrix}{{C^{R}(U)} = {{e(U)} - \left( {{d(U)} - \underset{\_}{d}} \right)}} \\{= {\left( {\alpha_{S} + {\beta_{S}{E\left\lbrack {U{U \geq k_{S - 1}}} \right\rbrack}} - \underset{\_}{d}} \right) - \left( {\alpha_{S} + {\beta_{S}U} - \underset{\_}{d}} \right)}} \\{= {\beta_{S}\left( {{E\left\lbrack {U{U \geq k_{S - 1}}} \right\rbrack} - U} \right)}}\end{matrix} & {11A{.1}{.11}}\end{matrix}$

Since the estimate that minimizes the variance is that random variable'sexpected value, the replicating portfolio is variance minimizing.Therefore, the theorem holds over this range.

Since cases 1, 2, and 3 cover the entire range of U, the generalreplication theorem holds for a buy of d. Using the definition of d inequation 11.2.1B and assumption 5, it is not hard to check that all ofthe a's are non-negative and at least one is zero.

11A.2: Proof for a Sell of the Derivatives Strategy d

To more easily distinguish sells of d from buys of d in this appendix,let ã_(s) denote the weight on the sth replication claim for a sell of dinstead of ã_(s) in the text.

Since (a₁, a₂, . . . , a_(2S−3), a_(2S−2)) is minimum variance for d−d,then ( d−d−a₁, d−d−a₂, . . . , d−d−a_(2S−3), d−d−a_(2S−2)) is minimumvariance for d−d. Comparing equation 11.2.3A-11.2.3D with11.2.3E-11.2.3H respectively

ã _(s) = d−d−a _(s) s=1, 2, . . . , 2S−2  11A.2.1

Therefore, (ã₁, ã₂, . . . , ã_(2S−3), ã_(2S−2)) is minimum variance ford−d. Using the definition of d in equation 11.2.1A and assumption 5, itis not hard to check that all of the ã's are non-negative and at leastone is zero. Thus, the replication weights for a sell of d are as smallas possible while remaining non-negative (non-negative replicationweights are a requirement to construct the DBAR equilibrium).

Appendix 11B: Derivation of the Self-Hedging Theorem of Section 11.4.5

Theorem: The self-hedging condition

$\begin{matrix}{{\sum\limits_{s = 1}^{{2S} - 2}{{d^{s}(U)}\left( {y_{s} + \frac{\theta_{s}}{p_{s}}} \right)}} = M} & {11B{.1}}\end{matrix}$

is equivalent to

$\begin{matrix}{{{y_{s} + \frac{\theta_{s}}{p_{s}}} = M}{{s = 1},2,\ldots \mspace{14mu},{{2S} - 2}}} & {11B{.2}}\end{matrix}$

for all values of U.

Proof: The three cases below divide the range of U into mutuallyexclusive and collectively exhaustive sets.

Case 1: U<k₁. The only replicating claim that pays out over this rangeis the first replicating instrument, the digital put struck at k₁.Therefore, the self-hedging condition 11B.1 combined with the definitionof d¹ from equation 11.2.2A is equivalent to

$\begin{matrix}{{y_{1} + \frac{\theta_{1}}{p_{1}}} = M} & {11B{.3}}\end{matrix}$

Thus the theorem holds over this range.

Case 2: k_(s−1)≦U<k_(s): s=2, 3, . . . , S−1. Over this range, the onlytwo claims that payout are the 2s−2^(nd) replicating claim and the2s−1^(st) replicating claim. In this case, the self-hedging condition11B.1 is equivalent to

$\begin{matrix}{{{{d^{{2s} - 2}(U)}\left( {y_{{2s} - 2} + \frac{\theta_{{2s} - 2}}{p_{{2s} - 2}}} \right)} + {{d^{{2s} - 1}(U)}\left( {y_{{2s} - 1} + \frac{\theta_{{2s} - 1}}{p_{{2s} - 1}}} \right)}} = M} & {11B{.4}}\end{matrix}$

Now, using the definition of d^(2s−2) from equation 11.2.2D and thedefinition of d^(2s-1) from equation 11.2.2E, equation 11B.4 becomes

$\begin{matrix}{{{\left( \frac{k_{s} - U}{k_{s} - k_{s - 1}} \right)\left( {y_{{2s} - 2} + \frac{\theta_{{2s} - 2}}{p_{{2s} - 2}}} \right)} + {\left( \frac{U - k_{s - 1}}{k_{s} - k_{s - 1}} \right)\left( {y_{{2s} - 1} + \frac{\theta_{{2s} - 1}}{p_{{2s} - 1}}} \right)}} = M} & {11B{.5}}\end{matrix}$

Next, set U equal to k_(s−1). Then, equation 11B.5 becomes

$\begin{matrix}{{{\left( \frac{k_{s} - k_{s - 1}}{k_{s} - k_{s - 1}} \right)\left( {y_{{2s} - 2} + \frac{\theta_{{2s} - 2}}{p_{{2s} - 2}}} \right)} + {\left( \frac{k_{s - 1} - k_{s - 1}}{k_{s} - k_{s - 1}} \right)\left( {y_{{2s} - 1} + \frac{\theta_{{2s} - 1}}{p_{{2s} - 1}}} \right)}} = M} & {11B{.6}}\end{matrix}$

Simplifying 11B.6 gives

$\begin{matrix}{{y_{{2s} - 2} + \frac{\theta_{{2s} - 2}}{p_{{2s} - 2}}} = M} & {11B{.7}}\end{matrix}$

Note that the left hand side of equation 11B.5 is a linear function of Uand note that the right hand side of equation 11B.5 is a constant M. Theleft hand side and the right hand side are equal for all values of Uover the range k_(s−1)≦U<k_(s) by the self-hedging condition. There areat least two values of U (since assumption 1 holds, i.e.,k_(s)−k_(s−1)≧2ρ) over this range. Since the linear function is equalfor at least two different values of U, it must be the case that thelinear function has a slope of zero (In symbols, α+βu₁=α+βu₂ impliesthat β=0 if u₁≠u₂). From equation 11B.5, note that the slope of thelinear function of U is

$\begin{matrix}{\left( {y_{{2s} - 1} + \frac{\theta_{{2s} - 1}}{p_{{2s} - 1}}} \right) - \left( {y_{{2s} - 2} + \frac{\theta_{{2s} - 2}}{p_{{2s} - 2}}} \right)} & {11B{.8}}\end{matrix}$

Therefore, the slope equally zero means that

$\begin{matrix}{{\left( {y_{{2s} - 1} + \frac{\theta_{{2s} - 1}}{p_{{2s} - 1}}} \right) - \left( {y_{{2s} - 2} + \frac{\theta_{{2s} - 2}}{p_{{2s} - 2}}} \right)} = 0} & {11B{.9}}\end{matrix}$

Substituting 11B.7 into 11B.9 yields that

$\begin{matrix}{{\left( {y_{{2s} - 1} + \frac{\theta_{{2s} - 1}}{p_{{2s} - 1}}} \right) - M} = 0} & {11B{.10}}\end{matrix}$

which implies that

$\begin{matrix}{{y_{{2s} - 1} + \frac{\theta_{{2s} - 1}}{p_{{2s} - 1}}} = M} & {11B{.11}}\end{matrix}$

Thus the theorem holds over this range.

Case 3: k_(S−1)≦U. The only replicating instrument that pays out overthis range is the 2S−2^(nd) replicating instrument, the digital callstruck at k_(S−1). Therefore, the self-hedging condition 11B.1 combinedwith the definition of d^(2S−2) from equation 11.2.211 is equivalent to

$\begin{matrix}{{y_{{2S} - 2} + \frac{\theta_{{2S} - 2}}{p_{{2S} - 2}}} = M} & {11B{.12}}\end{matrix}$

Thus the theorem holds over this range.

Since all three cases cover the entire range of U, this concludes theproof.

Appendix 11C: Probability Weighted Statistics from Sections 11.5.2 and11.5.3

This appendix discusses how to compute the probability weightedstatistics in tables 11.5.2E and 11.5.3E.

For the digital replicating basis in section 11.5.2, note that theoutcome of 59 is in state s=3 as shown in table 11.5.2A. In this case,using equation 11.5.2E

$\begin{matrix}\begin{matrix}{{\Pr \left\lbrack {U = 59} \right\rbrack} = \frac{p_{3}\rho}{k_{3} - k_{2}}} \\{= \frac{(0.134400500)(0.01)}{60 - 50}} \\{= 0.000134400500}\end{matrix} & {11C{.1}}\end{matrix}$

For the vanilla replicating basis in section 11.5.3, using equation11.5.3D, the probability is given by

$\begin{matrix}\begin{matrix}{{\Pr \left\lbrack {U = 59} \right\rbrack} = \frac{\left( {p_{4} + p_{5}} \right)\rho}{k_{3} - k_{2}}} \\{= \frac{\left( {0.000050029 + 0.299950032} \right) \times (0.01)}{60 - 50}} \\{= 0.0003000}\end{matrix} & {11C{.2}}\end{matrix}$

Based on these probabilities, summary measures of a statistic C can becomputed as follows

$\begin{matrix}{{Average} = {\sum\limits_{u}{{\Pr \left\lbrack {U = u} \right\rbrack}{C(u)}}}} & {11C{.3}} \\{{{Standard}\mspace{14mu} {Deviation}} = \sqrt{\sum\limits_{u}{{\Pr \left\lbrack {U = u} \right\rbrack}\left( {{C(u)} - {Average}} \right)^{2}}}} & {11C{.4}} \\{{{Semi}\text{-}{Standard}\mspace{14mu} {Deviation}} = \frac{\sqrt{\sum\limits_{{C{(u)}} < {Average}}{{\Pr \left\lbrack {U = u} \right\rbrack}\left( {{C(u)} - {Average}} \right)^{2}}}}{{\Pr \left\lbrack {{C(u)} < {Average}} \right\rbrack} + \frac{\Pr \left\lbrack {{C(u)} = {Average}} \right\rbrack}{2}}} & {11C{.5}} \\{{Skewness} = \frac{\sum\limits_{u}{{\Pr \left\lbrack {U = u} \right\rbrack}\left( {{C(u)} - {Average}} \right)^{3}}}{\left( {{Standard}\mspace{14mu} {Deviation}} \right)^{3}}} & {11C{.6}}\end{matrix}$

These formulas are used to compute the quantities in tables 11.5.2E and11.5.3E.

Appendix 11D: Notation Used in the Body of Text

a_(s): a scalar representing the replication weight for the sthreplication claim for strategy d;a_(j,s): a scalar representing the replication weight for thederivatives strategy d_(j) for replicating claim s for customer order j,j=1, 2, . . . , J;C^(R)(U): a function representing the replication P&L for the auctionsponsor based on the underlying U;C^(T)(U): a function representing the outcome dependent P&L for theauction sponsor based on the underlying U;

C^(θ)(U): a function representing the opening order P&L for the auctionsponsor based on the underlying U;

d and d(U): functions representing the payout on a European stylederivatives strategy based on the underlying U;d_(j) and d_(j)(U): functions representing the payout on a Europeanstyle derivatives strategy for customer order j, j=1, 2, . . . , J;d^(s) and d^(s)(U): functions representing the payout on the sthreplicating claim;d, d′, and d″: scalars representing typically the minimum payout of thederivatives strategy d;d, d′, and d″: scalars representing typically the maximum payout of thederivatives strategy d;d_(j) ′ and d_(j) ″: scalars representing typically the minimum payoutof the derivatives strategy d_(j), j=1, 2, . . . , J;d_(j) ′ and d_(j) ″: scalars representing typically the maximum payoutof the derivatives strategy d_(i), j=1, 2, . . . , J;E: the expectation operator;e(U): a function representing the payout on the replicating portfolio;Exp: the exponential function raising the argument to the power of e;G: a scalar representing the number of observations for an OLSregression;I: the indicator function;int: a function representing the greatest integer less than or equal tothe function's argument;j: a scalar used to index the customer orders in an auction j=1, 2, . .. , J;J: a scalar representing the total number of customer orders in anauction;k₁, k₂, k_(S−1): scalars representing strikes of the derivativesstrategies that customers can trade in an auction;k₀: a scalar representing the lower bound of U;k_(S): a scalar representing the upper bound of U;m_(s): a scalar representing the total filled premium for vanillareplicating claim s, s=1, 2, . . . , 2S−2;M: a scalar representing the total cleared premium in an auction;N: the cumulative distribution function for the standard normal;p_(s): a scalar representing the equilibrium price of the sthreplicating claim;Pr: the probability operator;r_(j): a scalar representing the requested number of contracts forcustomer order j, j=1, 2, . . . , J;

s: a scalar used to index across strikes or replication claims. Forstrikes s=1, 2, . . . , S−1. For digital replicating claims s=1, 2, . .. , S. For vanilla replicating claims s=1, 2, . . . , 2S−2. Foraugmented vanilla replicating claims s=1, 2, . . . , 2S;

S: a scalar representing the number of strikes plus one;U: a random variable representing the outcome of the underlying;u: a scalar representing a possible outcome of U;u_(g): a scalar representing a possible outcome of U, G;v: a scalar representing a specific strike or a specific replicationclaim;w: a scalar representing a specific strike;w_(j): scalar representing the limit price for customer order j, j=1, 2,. . . , J;w_(j) ^(a): a scalar representing the adjusted limit price for customerorder j,j=1, 2, . . . , J;x_(j): a scalar representing the equilibrium number of filled contractsfor customer order j, j=1, 2,x_(g,s) ^(OLS): a scalar representing an independent variable OLSregression for g=1, 2, . . . , G and s=1, 2, . . . , 2S−2;y_(s): a scalar representing the aggregate replicated customer payoutfor vanilla replicating claim s for s=1, 2, . . . , 2S−2;y_(g) ^(OLS): a scalar representing an explanatory variable in an OLSregression for g=1, 2, . . . , G;α_(s): a scalar representing the intercept of the payout function dbetween k_(s−1) and k_(s) for s=1, 2, . . . , S;β_(s): a scalar representing the slope of the payout function d betweenk_(s−1) and k_(s) for s=1, 2, . . . , S;ε_(g): a scalar representing the gth residual in a regression for g=1,2, . . . , G;θ_(s): a scalar representing the initial invested premium amount or theopening order premium amount for replicating claim s;Θ: a scalar representing the total amount of opening orders in anauction;μ: a scalar representing the mean of a normally distributed randomvariable;π_(j): a scalar representing the equilibrium price of customer order j,j=1, 2, . . . , J;π^(cf): a scalar representing the equilibrium price of a collaredforward;π^(ƒ): a scalar representing the equilibrium price of a forward;ρ: a scalar representing a measurable unit of the underlying U, whichcan be set at the level of precision to which the underlying U isreported or rounded by the auction sponsor;σ: a scalar representing the standard deviation of a normallydistributed random variable.

12. DETAILED DESCRIPTION OF THE DRAWINGS IN FIGS. 1 TO 28

Referring now to the drawings, similar components appearing in differentdrawings are identified by the same reference numbers.

FIGS. 1 and 2 show schematically a preferred embodiment of a networkarchitecture for any of the embodiments of a demand-based market orauction or DBAR contingent claims exchange (including digital options).As depicted in FIG. 1 and FIG. 2, the architecture conforms to adistributed Internet-based architecture using object oriented principlesuseful in carrying out the methods of the present invention.

In FIG. 1, a central controller 100 has a plurality software andhardware components and is embodied as a mainframe computer or aplurality of workstations. The central controller 100 is preferablylocated in a facility that has back-up power, disaster-recoverycapabilities, and other similar infrastructure, and is connected viatelecommunications links 110 with computers and devices 160, 170, 180,190, and 200 of traders and investors in groups of DBAR contingentclaims of the present invention. Signals transmitted usingtelecommunications links 110, can be encrypted using such algorithms asBlowfish and other forms of public and private key encryption. Thetelecommunications links 110 can be a dialup connection via a standardmodem 120; a dedicated line connection establishing a local area network(LAN) or wide area network (WAN) 130 running, for example, the Ethernetnetwork protocol; a public Internet connection 140; or wireless orcellular connection 150. Any of the computers and devices 160, 170, 180,190 and 200, depicted in FIG. 1, can be connected using any of the links120, 130, 140 and 150 as depicted in hub 111. Other telecommunicationslinks, such as radio transmission, are known to those of skill in theart.

As depicted in FIG. 1, to establish telecommunications connections withthe central controller 100, a trader or investor can use workstations160 running, for example, UNIX, Windows NT, Linux, or other operatingsystems. In preferred embodiments, the computers used by traders orinvestors include basic input/output capability, can include a harddrive or other mass storage device, a central processor (e.g., anIntel-made Pentium III processor), random-access memory, networkinterface cards, and telecommunications access. A trader or investor canalso use a mobile laptop computer 180, or network computer 190 having,for example, minimal memory and storage capability 190, or personaldigital assistant 200 such as a Palm Pilot. Cellular phones or othernetwork devices may also be used to process and display information fromand communicate with the central controller 100.

FIG. 2 depicts a preferred embodiment of the central controller 100comprising a plurality of software and hardware components. Computerscomprising the central controller 100 are preferably high-endworkstations with resources capable of running business operatingsystems and applications, such as UNIX, Windows NT, SQL Server, andTransaction Server. In a preferred embodiment, these computers arehigh-end personal computers with Intel-made (x86 “instruction set”)CPUs, at least 128 megabytes of RAM, and several gigabytes of hard drivedata storage space. In preferred embodiments, computers depicted in FIG.2 are equipped with JAVA virtual machines, thereby enabling theprocessing of JAVA instructions. Other preferred embodiments of thecentral controller 100 may not require the use of JAVA instruction sets.

In a preferred embodiment of central controller 100 depicted in FIG. 2,a workstation software application server 210, such as the WeblogicServer available from BEA Systems, receives information viatelecommunications links 110 from investors' computers and devices 160,170, 180, 190 and 200. The software application server 210 isresponsible for presenting human-readable user interfaces to investors'computers and devices, for processing requests for services frominvestors' computers and devices, and for routing the requests forservices to other hardware and software components in the centralcontroller 100. The user interfaces that can be available on thesoftware application server 210 include hypertext markup language (HTML)pages, JAVA applets and servlets, JAVA or Active Server pages, or otherforms of network-based graphical user interfaces known to those of skillin the art. For example, investors or traders connected via an Internetconnection for HTML can submit requests to the software applicationserver 210 via the Remote Method Invocation (RMI) and/or the InternetInter-Orb Protocol (HOP) running on top of the standard TCP/IP protocol.Other methods are known to those of skill in the art for transmittinginvestors' requests and instructions and presenting human readableinterfaces from the application server 210 to the traders and investors.For example, the software application server 210 may host Active ServerPages and communicate with traders and investors using DCOM.

In a preferred embodiment, the user interfaces deployed by the softwareapplication server 210 present login, account management, trading,market data, and other input/output information necessary for theoperation of a system for investing in replicated derivativesstrategies, financial products and/or groups of DBAR contingent claimsaccording to the present invention. A preferred embodiment uses the HTMLand JAVA applet/servlet interface. The HTML pages can be supplementedwith embedded applications or “applets” using JAVA based or ActiveXstandards or another suitable application, as known to one of skill inthe art.

In a preferred embodiment, the software application server 210 relies onnetwork-connected services with other computers within the centralcontroller 100. The computers comprising the central controller 100preferably reside on the same local area network (e.g., Ethernet LAN)but can be remotely connected over Internet, dedicated, dialup, or othersimilar connections. In preferred embodiments, networkintercommunication among the computers comprising central controller 100can be implemented using DCOM, CORBA, or TCP/IP or other stack servicesknown to those of skilled in the art.

Representative requests for services from the investors' computers tothe software application server 210 include: (1) requests for HTML pages(e.g., navigating and searching a web site); (2) logging onto the systemfor trading replicated derivatives strategies, replicated financialproducts, and/or DBAR contingent claims; (3) viewing real-time andhistorical market data and market news; (4) requesting analyticalcalculations such as returns, market risk, and credit risk; (5) choosinga derivatives strategy, financial product or group of DBAR contingentclaims of interest by navigating HTML pages and activating JAVA applets;(6) making an investment in a derivatives strategy, financial productsor one or more defined states of a group of DBAR contingent claims; and(7) monitoring investments in derivatives strategies, financial productsand groups of DBAR contingent claims.

In a preferred embodiment depicted in FIG. 2, an Object Request Broker(ORB) 230 can be a workstation computer operating specialized softwarefor receiving, aggregating, and marshalling service requests from thesoftware application server 210. For example, the ORB 230 can operate asoftware product called Visibroker, available from Inprise, and relatedsoftware products that provide a number of functions and servicesaccording to the Common Object Request Broker Architecture (CORBA)standard. In a preferred embodiment, one function of the ORB 230 is toprovide what are commonly known in the object-oriented software industryas directory services, which correlate computer code organized intoclass modules, known as “objects,” with names used to access thoseobjects. When an object is accessed in the form of a request by name,the object is instantiated (i.e., caused to run) by the ORB 230. Forexample, in a preferred embodiment, computer code organized into a JAVAclass module for the purpose of computing returns using a canonical DRFis an object named “DRF_Returns,” and the directory services of the ORB230 would be responsible for invoking this object by this name wheneverthe application server 210 issues a request that returns be computed.Similarly, in the case of DBAR digital options, computer code organizedinto a JAVA class module for the purpose of computing investment amountsusing a canonical DRF is an object named “OPF_Prices,” and the directoryservices of the ORB 230 would also be responsible for invoking thisobject by this name whenever the application server 210 issues a requestthat prices or investment amounts be computed.

In a preferred embodiment, another function of the ORB 230 is tomaintain what is commonly known in the object-oriented software industryas an interface repository, which contains a database of objectinterfaces. The object interfaces contain information regarding whichcode modules perform which functions. For example, in a preferredembodiment, a part of the interface of the object named “DRF_Returns” isa function which fetches the amount currently invested across thedistribution of states for a group of DBAR contingent claims. Similarly,for DBAR digital options, a part of the interface of the object named“OPF_Prices” is a function which fetches the requested payout orreturns, the selected outcomes and the limit prices or amounts for eachin a group of DBAR digital, options.

In a preferred embodiment, as in the other embodiments of the presentinvention, another function of the ORB 230 is to manage the length ofruntime for objects which are instantiated by the ORB 230, and to manageother functions such as whether the objects are shared and how theobjects manage memory. For example, in a preferred embodiment, the ORB230 determines, depending upon the request from the software applicationserver 210, whether an object which processes market data will sharesuch data with other objects, such as objects that allocate returns toinvestments in defined states.

In a preferred embodiment, as the other embodiments of the presentinvention, another function of the ORB 230 is to provide the ability forobjects to communicate asynchronously by responding to messages or dataat varying times and frequencies based upon the activity of otherobjects. For example, in a preferred embodiment, an object that computesreturns for a group of DBAR contingent claims responds asynchronously inreal-time to a new investment and recalculates returns automaticallywithout a request by the software application server 210 or any otherobject. In preferred embodiments, such asynchronous processes areimportant where computations in real-time are made in response to otheractivity in the system, such as a trader making a new investment or thefulfillment of the predetermined termination criteria for a group ofDBAR contingent claims.

In a preferred embodiment, as the other embodiments of the presentinvention, another function of the ORB 230 is to provide functionsrelated to what is commonly known in the object-oriented softwareindustry as marshalling. Marshalling in general is the process ofobtaining for an object the relevant data it needs to perform itsdesignated function. In preferred embodiments of the present invention,such data includes for example, trader and account information and canitself be manipulated in the form of an object, as is common in thepractice of object-oriented programming. Other functions and servicesmay be provided by the ORB 230, such as the functions and servicesprovided by the Visibroker product, according to the standards andpractices of the object-oriented software industry or as known to thoseof skill in the art.

In a preferred embodiment depicted in FIG. 2, which can be applied tothe other embodiments of the present invention, transaction server 240is a computer running specialized software for performing various tasksincluding: (1) responding to data requests from the ORB 230, e.g., user,account, trade data and market data requests; (2) performing relevantcomputations concerning groups of DBAR contingent claims, such asintra-trading period and end-of-trading-period returns allocations andcredit risk exposures; and (3) updating investor accounts based upon DRFpayouts for groups of DBAR contingent claims and applying debits orcredits for trader margin and positive outstanding investment balances.The transaction server 240 preferably processes all requests from theORB 230 and, for those requests that require stored data (e.g., investorand account information), queries data storage devices 260. In apreferred embodiment depicted in FIG. 2, a market data feed 270 suppliesreal-time and historical market data, market news, and corporate actiondata, for the purpose of ascertaining event outcomes and updatingtrading period returns. The specialized software running on transactionserver 240 preferably incorporates the use of object oriented techniquesand principles available with computer languages such as C++ or Java forimplementing the above-listed tasks.

As depicted in FIG. 2, in a preferred embodiment the data storagedevices 260 can operate relational database software such as Microsoft'sSQL Server or Oracle's 8i Enterprise Server. The types of databaseswithin the data storage devices 260 that can be used to support the DBARcontingent claim and exchange preferably comprise: (1) Trader andAccount databases 261; (2) Market Returns databases 262; (3) Market Datadatabases 263; (4) Event Data databases 264; (5) Risk databases 265; (6)Trade Blotter databases 266; and (7) Contingent Claims Terms andConditions databases 267. The kinds of data preferably stored in eachdatabase are shown in more detail in FIG. 4. In a preferred embodiment,connectivity between data storage devices 260 and transaction server 240is accomplished via TCP/IP and standard Database Connectivity Protocols(DBC) such as the JAVA DBC (JDBC). Other systems and protocols for suchconnectivity are known to those of skill in the art.

In reference to FIG. 2, application server 210 and ORB 230 may beconsidered to form an interface processor, while transaction server 240forms a demand-based transaction processor. Further, the databaseshosted on data storage devices 260 may be considered to form a tradestatus database. Investors, also referred to as traders, communicatingvia telecommunications links 110 from computers and devices 160, 170,180, 190, and 200, may be considered to perform a series of demand-basedinteractions, also referred to as demand-based transactions, with thedemand-based transaction processor. A series of demand-basedtransactions may be used by a trader, for example, to obtain marketdata, to establish a trade, or to close out a trade.

FIG. 3 depicts a preferred embodiment of the implementation of a groupof DBAR contingent claims. As depicted in FIG. 3, an exchange firstselects an event of economic significance 300. In the preferredembodiment, the exchange then partitions the possible outcomes for theevent into mutually exclusive and collectively exhaustive states 305,such that one state among the possible states in the partitioneddistribution is guaranteed to occur, and the sum of probabilities of theoccurrence of each partitioned state is unity. Trading can then commencewith the beginning 311 of the first trading period 310. In the preferredembodiment depicted in FIG. 3, a group of DBAR contingent claims hastrading periods 310, 320, 330, and 340, with trading period start date311, 321, 331, 341 respectively, followed by a predetermined timeinterval by each trading period's respective trading end dates 313, 323,333 and 343. The predetermined time interval is preferably of shortduration in order to attain continuity. In the preferred embodiment,during each trading period the transaction server 240 running JAVA codeimplementing the DRF for the group of DBAR contingent claims adjustsreturns immediately in response to changes in the amounts invested ineach of the defined states. Changes in market conditions during atrading period, such as price and volatility changes, as well as changesin investor risk preferences and liquidity conditions in the underlyingmarket, among other factors, will cause amounts invested in each definedstate to change thereby reflecting changes in expectations of tradersover the distribution of states defining the group of DBAR contingentclaims.

In a preferred embodiment, the adjusted returns calculated during atrading period, i.e., intra-trading period returns, are of informationalvalue only—only the returns which are finalized at the end of eachtrading period are used to allocate gains and losses for a trader'sinvestments in a group or portfolio of groups of DBAR contingent claims.In a preferred embodiment, at the end of each trading period, forexample, at trading end dates 313, 323, 333, and 343, finalized returnsare allocated and locked in. The finalized returns are the rates ofreturn to be allocated per unit of amount invested in each defined stateshould that state occur. In a preferred embodiment, each trading periodcan therefore have a different set of finalized returns as marketconditions change, thereby enabling traders to make investments duringlater trading periods which hedge investments from earlier tradingperiods that have since closed.

In another preferred embodiment, not depicted, trading periods overlapso that more than one trading period is open for investment on the sameset of predefined states. For example, an earlier trading period canremain open while a subsequent trading period opens and closes. Otherpermutations of overlapping trading periods are possible and areapparent to one of skill in the art from this specification or practiceof the present invention.

The canonical DRF, as previously described, is a preferred embodiment ofa DRF which takes investment across the distribution of states and eachstate, the transaction fee, and the event outcome and allocates a returnfor each state if it should occur. A canonical DRF of the presentinvention, as previously described, reallocates all amounts invested instates that did not occur to the state that did occur. Each trader thathas made an investment in the state that did occur receives a pro-ratashare of the trades from the non-occurring states in addition to theamount he originally invested in the occurring state, less the exchangefee. In the preferred embodiment depicted in FIG. 3, at the close of thefinal trading period 343, trading ceases and the outcome for the eventunderlying the contingent claim is determined at close of observationperiod 350. In a preferred embodiment, only the outcome of the eventunderlying the group of contingent claims must be uncertain during thetrading periods while returns are being locked in. In other words, theevent underlying the contingent claims may actually have occurred beforethe end of trading so long as the actual outcome remains unknown, forexample, due to the time lag in measuring or ascertaining the event'soutcome. This could be the case, for instance, with macroeconomicstatistics like consumer price inflation.

In the preferred embodiment depicted in FIG. 2, once the outcome isobserved at time 350, process 360 operates on the finalized returns fromall the trading periods and determines the payouts. In the case of acanonical DRF previously described, the amounts invested in the losinginvestments finance the payouts to the successful investments, less theexchange fee. In a canonical DRF, successful investments are those madeduring a trading period in a state which occurred as determined at time350, and unsuccessful investments are those made in states which did notoccur. Examples 3.1.1-3.1.25 above illustrate various preferredembodiments of a group of DBAR contingent claims using a canonical DRF.In the preferred embodiment depicted in FIG. 3, the results of process360 are made available to traders by posting the results for all tradingperiods on display 370. In a preferred embodiment not depicted, traderaccounts are subsequently updated to reflect these results.

FIG. 4 provides a more detailed depiction of the data storage devices260 of a preferred embodiment of a DBAR contingent claims exchange whichcan be applied to the other embodiments of the present invention. In apreferred embodiment, data storage devices 260, on which relationaldatabase software is installed as described above, is a non-volatilehard drive data storage system, which may comprise a single device ormedium, or may be distributed across a plurality of physical devices,such as a cluster of workstation computers operating relational databasesoftware, as described previously and as known in the art. In apreferred embodiment, the relational database software operating on thedata storage devices 260 comprises relational database tables, storedprocedures, and other database entities and objects that are commonlycontained in relational database software packages. In the preferredembodiment depicted in FIG. 4, databases 261-267 each contain suchtables and other relational database entities and objects necessary ordesirable to implement an embodiment of the present invention. FIG. 4identifies the kinds of information that can be stored in such devices.Of course, the kinds of data shown in the drawing are not exhaustive.The storage of other data on the same or additional databases may beuseful depending on the nature of the contingent claim being traded.Moreover, in the preferred embodiment depicted in FIG. 4, certain dataare shown in FIG. 4 as stored in more than one storage device. Invarious other preferred embodiments, such data may be stored in only onesuch device or may be calculated. Other database designs andarchitectures will be apparent to those of skill in the art from thisspecification or practice of the present invention.

In the preferred embodiment depicted in FIG. 4, the Trader and Accountdatabase 261 stores data related to the identification of a DBAR tradersuch as name, password, address, trader identification number, etc. Datarelated to the trader's credit rating can also be stored and updated inresponse to changes in the trader's credit status. Other informationthat can be stored in Trader and Account database 261 includes datarelated to the trader's account, for example, active and inactiveinvestments, the trader's balance, the trader's margin limits,outstanding margin amounts, interest credited on outstanding tradebalances and interest paid on outstanding margin balances, anyrestrictions the trader may have regarding access to his account, andthe trader's profit and loss information regarding active and inactiveinvestments. Information related to multi-state investments to beallocated can also be stored in Trader and Account database 261. Thedata stored in database 261 can be used, for example, to issue accountrelated statements to traders.

In the preferred embodiment depicted in FIG. 4, the Market Returnsdatabase 262 contains information related to returns available atvarious times for active and inactive groups of DBAR contingent claims.In a preferred embodiment, each group of contingent claims in database262 is identified using a unique identifier previously assigned to thatgroup. Returns for each defined state for each group of contingentclaims reflected are stored in database 262. Returns calculated andavailable for display to traders during a given trading period arestored in database 262 for each state and for each claim. At the end ofeach trading period, finalized returns are computed and stored in MarketReturns database 262. Marginal returns, as previously described, canalso be stored in database 262. The data in Market Returns database 262may also include information relevant to a trader's decisions such ascurrent and past intra-period returns, as well as information used todetermine payouts by a DRF or investment amounts by an OPF for a groupof DBAR contingent claims.

In the preferred embodiment depicted in FIG. 4, Market Data database 263stores market data from market data feed 270. In a preferred embodiment,the data in Market Data database 263 include data relevant for the typesof contingent claims that can be traded on a particular exchange. In apreferred embodiment, real-time market data include data such asreal-time prices, yields, index levels, and other similar information.In a preferred embodiment, such real-time data from Market Data database263 are presented to traders to aid in making investment decisions canbe used by the DRF to allocate returns and by the OPF to determineinvestment amounts for groups of contingent claims that depend on suchinformation. Historical data relating to relevant groups of DBARcontingent claims can also be stored in Market Data database 263. Inpreferred embodiments, news items related to underlying groups of DBARcontingent claims (e.g., comments by the Federal Reserve) are alsostored in Market Data database 263 and can be retrieved by traders.

In the preferred embodiment depicted in FIG. 4, Event Data database 264stores data related to events underlying the groups of DBAR contingentclaims that can be traded on an exchange. In a preferred embodiment,each event is identified by a previously assigned event identificationnumber. Each event has one or more associated group of DBAR contingentclaims based on that event and is so identified with a previouslyassigned contingent claim group identification number. The type of eventcan also be stored in Event database 264, for example, whether the eventis based on a closing price of a security, a corporate earningsannouncement, a previously calculated but yet to be released economicstatistic, etc. The source of data used to determine the outcome of theevent can also be stored in Event database 264. After an event outcomebecomes known, it can also be stored in Event database 264 along withthe defined state of the respective group of contingent claimscorresponding to tht outcome.

In the preferred embodiment depicted in FIG. 4, Risk database 265 storesthe data and results and analyses related to the estimation andcalculation of market risk and credit risk. In a preferred embodiment,Risk database 265 correlates the derived results with an accountidentification number. The market and credit risk quantities that can bestored are those related to the calculation of CAR and CCAR, such as thestandard deviation of unit returns for each state, the standarddeviation of dollar returns for each state, the standard deviation ofdollar returns for a given contingent claim, and portfolio CAR.Intermediate estimation and simulation data such as correlation matricesused in VAR-based CAR and CCAR calculations and scenarios used inMCS-based calculations can also be stored in Risk database 265.

In the preferred embodiment depicted in FIG. 4, Trade Blotter database266 contains data related to the investments, both active and inactive,made by traders for all the groups of DBAR contingent claims (as well asderivatives strategies, financial products) that can be traded on theparticular exchange. Such data may include previously assigned traderidentification numbers previously assigned investment identificationnumbers, previously assigned account identification numbers, previouslyassigned contingent claim identification numbers, state identificationnumbers previously assigned corresponding to each defined state, thetime of each investment, the units of value used to make eachinvestments (e.g., dollars), the investment amounts, the desired orrequested payouts or returns, the limits on investment amounts (for DBARdigital options), how much margin is used to make the investments, andpreviously assigned trading period identification numbers, as well aspreviously assigned derivatives strategy numbers and/or financialproducts (not shown). In addition, data related to whether an investmentis a multi-state investment can also be stored. The payout distributionthat a trader desires to replicate and that the exchange will implementusing a multi-state investment allocation, as described above, can alsobe stored in Trade Blotter database 266.

In the preferred embodiment depicted in FIG. 4, Contingent Claims Termsand Conditions database 267 stores data related to the definition andstructure of each group of DBAR contingent claims. In a preferredembodiment, such data are called “terms and conditions” to indicate thatthey relate to the contractual terms and conditions under which tradersagree to be bound, and roughly correspond to material found inprospectuses in traditional markets. In a preferred embodiment, as wellas other embodiments of the present invention, the terms and conditionsprovide the fundamental informkion regarding the nature of thecontingent claim to be traded, e.g., the number of trading periods, thetrading period(s)' start and end times, the type of event underlying thecontingent claim, how the DRF finances successful investments fromunsuccessful investments, how the OPF determines order prices orinvestment amounts as a function of the requested payout, selection ofoutcomes and limits for each order for a DBAR auction or market, thetime at which the event is observed for determining the outcome, otherpredetermined termination criteria, the partition of states in whichinvestments can be made, and the investment and payout value units(e.g., dollars, numbers of shares, ounces of gold, etc.). In a preferredembodiment, contingent claim and event identification numbers areassigned and stored in Contingent Claims Terms and Conditions database267 so that they may be readily referred to in other tables of the datastorage devices.

FIG. 5 shows a flow diagram depicting illustrative processes used andillustrative decisions made by a trader using a preferred embodiment ofthe present invention. For purposes of illustration in FIG. 5, it isassumed that the trader is making an investment in a DBAR rangederivative (RD) examples of which are disclosed above. In particular, itis assumed for the purposes of illustration that the DBAR RD investmentbeing made is in a contingent claim based upon the closing price of IBMcommon stock on Aug. 3, 1999 (as indicated in the display 501 of FIG.6).

In process 401, depicted in FIG. 5, the trader requests access to theDBAR contingent claim exchange. As previously described in a preferredembodiment, the software application server 210 (depicted in FIG. 2)processes this request and routes it to the ORB 230, which instantiatesan object responsible for the authentication of traders on the exchangeon transaction server 240. The authentication object on transactionserver 240, for example, queries the Trader and Account database 261(depicted in FIG. 4) for the trader's username, password, and otheridentifying information supplied. The authentication object responds byeither allowing or denying access as indicated in process 402 depictedin FIG. 5. If authentication fails in this illustration, process 403prompts the trader to retry a logon or establish valid credentials forlogging onto the system. If the trader has been granted access, thesoftware application server 210 (depicted in FIG. 2) will display to thetrader many user interfaces that may be of interest. For example, in apreferred embodiment, the trader can navigate through a sample of groupsof DBAR contingent claims currently being traded, as represented inprocess 404. The trader may also check current market conditions byrequesting those interfaces in process 404 that contain current marketdata as obtained from market data feed 270 (depicted in FIG. 2) andstored in Market Data database 263 (as depicted in FIG. 4). Process 405of FIG. 5 represents the trader requesting the application server 210for relevant information regarding the trader's account, such as thetrader's current portfolio of trades, trade amounts, current amount ofmargin outstanding, and account balances. In a preferred embodiment,this information is obtained by objects running on transaction server240 (FIG. 2) that query Trader and Account database 261 and TradeBlotter database 266 (FIG. 4).

As depicted in FIG. 5, process 407 represents the selection of a groupof DBAR contingent claims by a trader for the purpose of making aninvestment. The application server 210 (depicted in FIG. 2) can presentuser interfaces to the trader such as the interface shown in FIG. 6 asis known in the art. Process 408 represents the trader requesting dataand analysis which may include calculations as to the effect thetrader's proposed investment would have on the current returns. Thecalculations can be made using the implied “bid” and “offer” demandresponse equations described above. The processes that perform thesedata requests and manipulation of such data are, in a preferredembodiment, objects running on transaction server 240 (as depicted inFIG. 2). These objects, for example, obtain data from database 262 (FIG.4) by issuing a query that requests investment amounts across thedistribution of states for a given trading period for a given group ofcontingent claims. With the investment amount data, other objectsrunning on transaction server 240 (FIG. 2) can perform marginal returnscalculations using the DRF of the group of contingent claims asdescribed above. Such processes are objects managed by the ORB 230 (asdepicted in FIG. 2).

Returning to the illustration depicted in FIG. 5, process 411 representsa trader's decision to make an investment for a given amount in one ormore defined states of the group of DBAR contingent claims of interest.In a preferred embodiment, the trader's request to make an investmentidentifies the particular group of claims, the state or states in whichinvestments are to be made, the amount to be invested in the state orstates, and the amount of margin to be used, if any, for theinvestments.

Process 412 responds to any requests to make an investment on margin.The use of margin presents the risk that the exchange may not be able tocollect the entire amount of a losing investment. Therefore, inpreferred embodiments, an analysis is performed to determine the amountof risk to which a current trader is exposed in relation to the amountof margin loans the trader currently has outstanding. In process 413such an analysis is carried out in response to a margin request by thetrader.

The proposed trade or trades under consideration may have the effect ofhedging or reducing the total amount of risk associated with thetrader's active portfolio of investments in replicated derivativesstrategies, financial products, and groups of DBAR contingent claims.Accordingly, in a preferred embodiment, the proposed trades and marginamounts should be included in a CAR analysis of the trader's portfolio.

In a preferred embodiment, the CAR analysis performed by process 413,depicted in FIG. 5, can be conducted according to the VAR, MCS, or HSmethodologies previously discussed, using data stored in Risk database265 (FIG. 2), such as correlation of state returns, correlation ofunderlying events, etc. In a preferred embodiment, the results of theCAR calculation are also stored in Risk database 265. As depicted inFIG. 5, process 414 determines whether the trader has sufficient equitycapital in his account by comparing the computed CAR value and thetrader's equity in accordance with the exchange's margin rules. Inpreferred embodiments, the exchange requires that all traders maintain alevel of equity capital equal to some portion or multiple of the CARvalue for their portfolios. For example, assuming CAR is computed with a95% statistical confidence as described above, the exchange may requirethat traders have 10 times CAR as equity in their accounts. Such arequirement would mean that traders would suffer drawdowns to equity of10% approximately 5% of the time, which might be regarded as areasonable tradeoff between the benefits of extending margin to tradersto increase liquidity and the risks and costs associated with traderdefault. In addition, in preferred embodiments, the exchange can alsoperform CCAR calculations to determine the amount of credit risk in thegroup of DBAR contingent claims due to each trader. In a preferredembodiment, if a trader does not have adequate equity in his account orthe amount of credit risk posed by the trader is too great, the requestfor margin is denied, as depicted in process 432 (FIG. 5).

As further depicted in FIG. 5, if the trader has requested no margin orthe trader has passed the margin tests applied in process 414, process415 determines whether the investment is one to be made over multiplestates simultaneously in order to replicate a trader's desired payoutdistribution over such states. If the investment is multi-state, process460 requests trader to enter a desired payout distribution. Suchcommunication will comprise, for example, a list of constituent statesand desired payouts in the event that each constituent state occurs. Forexample, for a four-state group of DBAR contingent claims, the tradermight submit the four dimensional vector (10, 0, 5, 2) indicating thatthe trader would like to replicate a payout of 10 value units (e.g.,dollars) should state 1 occur, no payout should state 2 occur, 5 unitsshould state 3 occur, and 2 units should state 4 occur. In a preferredembodiment, this information is stored in Trade Blotter database 266(FIG. 4) where it will be available for the purposes of determining theinvestment amounts to be allocated among the constituent states for thepurposed of replicating the desired payouts. As depicted in FIG. 5, ifthe investment is a multi-state investment, process 417 makes aprovisional allocation of the proposed investment amount to each of theconstituent states.

As further depicted in FIG. 5, the investment details and information(e.g., contingent claim, investment amount, selected state, amount ofmargin, provisional allocation, etc.) are then displayed to the traderfor confirmation by process 416. Process 418 represents the trader'sdecision whether to make the investment as displayed. If the traderdecides against making the investment, it is not executed as representedby process 419. If the trader decides to make the investment and process420 determines that it is not a multi-state investment, the investmentis executed, and the trader's investment amount is recorded in therelevant defined state of the group of DBAR contingent claims accordingto the investment details previously accepted. In a preferredembodiment, the Trade Blotter database 266 (FIG. 4) is then updated byprocess 421 with the new investment information such as the trader ID,trade ID, account identification, the state or states in whichinvestments were made, the investment time, the amount invested, thecontingent claim identification, etc.

In the illustration depicted in FIG. 5, if the trader decides to makethe investment, and process 420 determines that it is a multi-stateinvestment, process 423 allocates the invested amount to the constituentstates comprising the multi-state investment in amounts that generatethe trader's desired payout distribution previously communicated to theexchange in process 460 and stored in Trader Blotter database 266 (FIG.4). For example, in a preferred embodiment, if the desired payouts areidentical payouts no matter which state occurs among the constituentstates, process 423 will update a suspense account entry and allocatethe multi-state trade in proportion to the amounts previously investedin the constituent states. Given the payout distribution previouslystored, the total amount to be invested, and the constituent states inwhich the “new” investment is to be made, then the amount to be investedin each constituent state can be calculated using the matrix formulaprovided in Example 3.1.21, for example. Since these calculations dependon the existing distributions of amounts invested both during and at theend of trading, in a preferred embodiment reallocations are performedwhenever the distribution of amounts invested (and hence returns)change.

As further depicted in FIG. 5, in response to a new investment, Process422 updates the returns for each state to reflect the new distributionof amounts invested across the defined states for the relevant group ofDBAR contingent claims. In particular, process 422 receives the newtrade information from Trade Blotter database 266 as updated by process421, if the investment is not multi-state, or from Trader and Accountdatabase 261 as updated by suspense account process 423, if theinvestment is a multi-state investment. Process 422 involves the ORB 230(FIG. 2) instantiating an object on transaction server 240 forcalculating returns in response to new trades. In this illustration, theobject queries the new trade data from the Trade Blotter database 266 orthe suspense account in Trader and Account database 261 (FIG. 4),computes the new returns using the DRF for the group of contingentclaims, and updates the intra-trading period returns stored in MarketReturns database 262.

As depicted in FIG. 5, if the investment is a multi-state investment asdetermined by process 450, the exchange continues to update the suspenseaccount to reflects the trader's desired payout distribution in responseto subsequent investments entering the exchange. Any updatedintra-trading period returns obtained from process 422 and stored inMarket Returns database 262 are used by process 423 to perform areallocation of multi-state investments to reflect the updated returns.If the trading period has not closed, as determined by process 452, thereallocated amounts obtained from the process 423 are used, along withinformation then simultaneously stored in Trade Blotter database 266(FIG. 4), to perform further intra-trading period update of returns, perprocess 422 shown in FIG. 5. However, if the trading period has closed,as determined in this illustration by process 452, then the multi-statereallocation is performed by process 425 so that the returns for thetrading period can be finalized per process 426.

In a preferred embodiment, the closing of the trading period is animportant point since at that point the DRF object running onTransaction server 240 (FIG. 2) calculates the finalized returns andthen updates Market Returns database 262 with those finalized returns,as represented by process 426 depicted in FIG. 5. The finalized returnsare those which are used to compute payouts once the outcome of theevent and, therefore, the state which occurred are known and all otherpredetermined termination criteria are fulfilled. Even though amulti-state reallocation process 425 is shown in FIG. 5 between process452 and process 426, multi-state reallocation process 425 is not carriedout if the investment is not a multi-state investment.

Continuing with the illustration depicted in FIG. 5, process 427represents the possible existence of subsequent trading periods for thesame event on which the given group of DBAR contingent claims is based.If such periods exist, traders may make investments during them, andeach subsequent trading period would have its own distinct set offinalized returns. For example, the trader in a group of contingentclaims may place a hedging investment in one or more of the subsequenttrading periods in response to changes in returns across the tradingperiods in accordance with the method discussed in Example 3.1.19 above.The ability to place hedging trades in successive trading periods, eachperiod having its own set of finalized returns, allows the trader tolock-in or realize profits and losses in virtually continuous time asreturns change across the trading periods. In a preferred embodiment,the plurality of steps represented by process 427 are performed aspreviously described for the earlier portions of FIG. 5.

As further depicted in FIG. 5, process 428 marks the end of all thetrading periods for a group of contingent claims. In a preferredembodiment, at the end of the last trading period, the Market Returnsdatabase 262 (FIG. 4) contains a set of finalized returns for eachtrading period of the group of contingent claims, and Trade Blotterdatabase 266 contains data on every investment made by every trader onthe illustrative group of DBAR contingent claims.

In FIG. 5, process 429 represents the observation period during whichthe outcome of the event underlying the contingent claim is observed,the occurring state of the DBAR contingent claim determined and anyother predetermined termination criteria are fulfilled. In a preferredembodiment, the event outcome is determined by query of the Market Datadatabase 263 (FIG. 4), which has been kept current by Market Data Feed270. For example, for a group of contingent claims on the event of theclosing price of IBM on Aug. 3, 1999, the Market Data database 263 willcontain the closing price, 119 3/8, as obtained from the specified eventdata source in Event Data database 264. The event data source might beBloomberg, in which case an object residing on transaction server 240previously instantiated by ORB 230 will have updated the Market Returnsdatabase 262 with the closing price from Bloomberg. Another similarlyinstantiated object on transaction server 240 will query the MarketReturns database 262 for the event outcome (119 3/8), will query theContingent Claims Terms and Conditions database 267 for the purpose ofdetermining the state identification corresponding to the event outcome(e.g., Contingent Claim # 1458, state #8) and update the event and stateoutcomes into the Event Data database 264.

As further depicted in FIG. 5, process 430 shows an object instantiatedon transaction server 240 by ORB 230 performing payout calculations inaccordance with the DRF and other terms and conditions as contained inContingent Claims Terms and Conditions database 267 for the given groupof contingent claims. In a preferred embodiment, the object isresponsible for calculating amounts to be paid to successful investmentsand amounts to be collected from unsuccessful investments, i.e.,investments in the occurring and non-occurring states, respectively.

As further depicted in FIG. 5, process 431 shows trader account datastored in Trader and Account database 261 (FIG. 4) being updated by theobject which determines the payouts in process 430. Additionally, inprocess 431 in this illustration and preferred embodiments, outstandingcredit and debit interest corresponding to positive and margin balancesare applied to the relevant accounts in Trader and Account database 261.

FIG. 6 depicts as preferred embodiment of a sample HTML page used bytraders in an exchange for groups of DBAR contingent claims whichillustrates sample display 500 with associated input/output devices,such as display buttons 504-507 and can be used with other embodimentsof the present invention. As depicted in FIG. 6, descriptive data 501illustrate the basic investment and market information relevant to aninvestment. In the investment illustrated in FIG. 6, the event is theclosing price of IBM common stock at 4:00 p.m. on Aug. 3, 1999. Asdepicted in FIG. 6, the sample HTML page displays amount invested ineach defined state, and returns available from Market Returns database262 depicted in FIG. 4. In this illustration and in preferredembodiments, returns are calculated on transaction server 240 (FIG. 2)using, for example, a canonical DRF. As also depicted in FIG. 6,real-time market data is displayed in an intraday “tick chart”,represented by display 503, using data obtained from Market Data Feed270, as depicted in FIG. 7, and processed by transaction server 240,depicted in FIG. 2. Market data may also be stored contemporaneously inMarket Data database 263.

In the preferred embodiment depicted in FIG. 6, traders may make aninvestment by selecting Trade button 504. Historical returns and timeseries data, from Market Data database 263 may be viewed by selectingDisplay button 505. Analytical tools for calculating opening orindicative returns or simulating market events are available by requestfrom Software Application Server 210 via ORB 230 and Transaction Server240 (depicted in FIG. 2) by selecting Analyze button 506 in FIG. 6. Asreturns change throughout the trading period, a trader may want todisplay how these returns have changed. As depicted in FIG. 6, theseintraday or intraperiod returns are available from Market Returnsdatabase 262 by selecting Intraday Returns button 507. In addition,marginal intra-period returns, as discussed previously, can be displayedusing the same data in Market Returns database 262 (FIG. 2). In apreferred embodiment, it is also possible for each trader to viewfinalized returns from Market Returns database 262. In preferredembodiments that are not depicted, display 500 also includes informationidentifying the group of contingent claims (such as the claim type andevent) available from the Contingent Claims Terms and Conditionsdatabase 267 or current returns available from Market Returns database262 (FIG. 2). In other preferred embodiments (e.g., any embodiments ofthe present invention), display 500 includes means for requesting otherservices which may be of interest to the trader, such as the calculationof marginal returns, for example by selecting

Intraday Returns button 507, or the viewing of historical data, forexample by selecting Historical Data button 505.

FIG. 7 depicts a preferred embodiment of the Market Data Feed 270 ofFIG. 2 in greater detail. In a preferred embodiment depicted in FIG. 7,which can be applied to other embodiments of the present invention,real-time data feed 600 comprises quotes of prices, yields, intradaytick graphs, and relevant market news and example sources. Historicaldata feed 610, which is used to supply market data database 263 withhistorical data, illustrates example sources for market time seriesdata, derived returns calculations from options pricing data, andinsurance claim data. Corporate action data feed 620 depicted in FIG. 7illustrates the types of discrete corporate-related data (e.g., earningsannouncements, credit downgrades) and their example sources which canform the basis for trading in groups of DBAR contingent claims of thepresent invention. In preferred embodiments, functions listed in process630 are implemented on transaction server 240 (FIG. 2) which takesinformation from data feeds 600, 610, and 620 for the purposes ofallocating returns, simulating outcomes, calculating risk, anddetermining event outcomes (as well as for the purpose of determininginvestment amounts).

FIG. 8 depicts a preferred embodiment of an illustrative graph ofimplied liquidity effects of investments in a group of DBAR contingentclaims. As discussed above, in preferred embodiments of the presentinvention, liquidity variations within a group of DBAR contingent claimimpose few if any costs on traders since only the finalized or closingreturns for a trading period matter to a trader's return. This contrastswith traditional financial markets, in which local liquidity variationsmay result in execution of trades at prices that do not fairly representfair fundamental value, and may therefore impose permanent costs ontraders.

Liquidity effects from investments in groups of DBAR contingent claims,as illustrated in FIG. 8, include those that occur when an investmentmaterially and permanently affects the distribution of returns acrossthe states. Returns would be materially and perhaps permanently affectedby a trader's investment if, for example, very close to the tradingperiod end time, a trader invested an amount in a state that representeda substantial percentage of aggregate amount previously invested in thatstate. The curves depicted FIG. 8 show in preferred embodiments themaximum effect a trader's investment can have on the distribution ofreturns to the various states in the group of DBAR contingent claims.

As depicted in FIG. 8, the horizontal axis, p, is the amount of thetrader's investment expressed as a percentage of the total amountpreviously invested in the state (the trade could be a multi-stateinvestment, but a single state is assumed in this illustration). Therange of values on the horizontal axis depicted in FIG. 8 has a minimumof 0 (no amount invested) to 10% of the total amount invested in aparticular state. For example, assuming the total amount invested in agiven state is $100 million, the horizontal axis of FIG. 8 ranges from anew investment amount of 0 to $10 million.

The vertical axis of FIG. 8 represents the ratio of the impliedbid-offer spread to the implied probability of the state in which a newinvestment is to be made. In a preferred embodiment, the impliedbid-offer spread is computed as the difference between the implied“offer” demand response, q_(i) ^(O)(ΔT_(i)), and the implied “bid”demand response, q_(i) ^(B)(ΔT_(i)), as defined above. In other words,values along the vertical axis depicted in FIG. 8 are defined by thefollowing ratio:

$\frac{{q_{i}^{O}\left( {\Delta \; T_{i}} \right)} - {q_{i}^{B}\left( {\Delta \; T_{i}} \right)}}{q_{i}}$

As displayed in FIG. 8, this ratio is computed using three differentlevels of q_(i), and the three corresponding lines for each level aredrawn over the range of values of p: the ratio is computed assuming alow implied q_(i) (q_(i)=0.091, denoted by the line marked S(p,l)), amiddle-valued q, (q_(i)=0.333, denoted by the line marked S(p,m)), and ahigh value for q_(i) (q_(i)=0.833 denoted by the line marked S(p,h)), asshown.

If a trader makes an investment in a group of DBAR contingent claims ofthe present invention and there is not enough time remaining in thetrading period for returns to adjust to a fair value, then FIG. 8provides a graphical depiction, in terms of the percentage of theimplied state probability, of the maximum effect a trader's owninvestment can have on the distribution of implied state probabilities.The three separate curves drawn correspond to a high demand and highimplied probability (S(p,h)), medium demand and medium impliedprobability (S(p,m)), and low demand and low implied probability(S(p,l)). As used in this context, the term “demand” means the amountpreviously invested in the particular state.

The graph depicted in FIG. 8 illustrates that the degree to which theamount of a trader's investment affects the existing distribution ofimplied probabilities (and hence returns) varies with the amount ofdemand for the existing state as well as the amount of the trader'sinvestment. If the distribution of implied probabilities is greatlyaffected, this corresponds to a larger implied bid-offer spread, asgraphed on the vertical axis of the graph of FIG. 8. For example, forany given investment amount p, expressed as a percentage of the existingdemand for a particular state, the effect of the new investment amountis largest when existing state demand is smallest (line S(p,l),corresponding to a low demand/low implied probability state). Bycontrast, the effect of the amount of the new investment is smallestwhen the existing state demand is greatest (S(p,h), corresponding to ahigh demand/high implied probability state). FIG. 8 also confirms that,in preferred embodiments, for all levels of existing state demand, theeffect of the amount invested on the existing distribution of impliedprobabilities increases as the amount to be invested increases.

FIG. 8 also illustrates two liquidity-related aspects of groups of DBARcontingent claims of the present invention. First, in contrast to thetraditional markets, in preferred embodiments of the present inventionthe effect of a trader's investment on the existing market can bemathematically determined and calculated and displayed to all traders.Second, as indicated by FIG. 8, the magnitude of such effects are quitereasonable. For example, in preferred embodiments as depicted by FIG. 8,over a wide range of investment amounts ranging up to several percent ofthe existing demand for a given state, the effects on the market of suchinvestments amounts are relatively small. If the market has time toadjust after such investments are added to demand for a state, theeffects on the market will be only transitory and there may be no effecton the implied distribution of probabilities owing to the trader'sinvestment. FIG. 8 illustrates a “worst case” scenario by implicitlyassuming that the market does not adjust after the investment is addedto the demand for the state.

FIGS. 9 a to 9 c illustrate, for a preferred embodiment of a group ofDBAR contingent claims, the trader and credit relationships and howcredit risk can be quantified, for example in process 413 of FIG. 5.FIG. 9 a depicts a counterparty relationship for a traditional swaptransaction, in which two counterparties have previously entered into a10-year swap which pays a semi-annual fixed swap rate of 7.50%. Thereceiving counterparty 701 of the swap transaction receives the fixedrate and pays a floating rate, while the paying counterparty 702 paysthe fixed rate and receives the floating rate. Assuming a $100 millionswap trade and a current market fixed swap rate of 7.40%, based uponwell-known swap valuation principles implemented in software packagessuch as are available from Sungard Data Systems, the receivingcounterparty 701 would receive a profit of $700,000 while the payingswap counterparty 702 would have a loss of $700,000. The receiving swapcounterparty 701 therefore has a credit risk exposure to the paying swapcounterparty 702 as a function of $700,000, because the arrangementdepends on the paying swap party 702 meeting its obligation.

FIG. 9 b depicts illustrative trader relationships in which a preferredembodiment of a group of the DBAR contingent claims and exchangeeffects, as a practical matter, relationships among all the traders. Asdepicted in FIG. 9 b, traders C1, C2, C3, C4, and C5 each have investedin one or more states of a group of DBAR contingent claims, with definedstates S1 to S8 respectively corresponding to ranges of possibleoutcomes for the 10 year swap rate, one year forward. In thisillustration, each of the traders has a credit risk exposure to all theothers in relation to the amount of each trader's investment, how muchof each investment is on margin, the probability of success of eachinvestment at any point in time, the credit quality of each trader, andthe correlation between and among the credit ratings of the traders.This information is readily available in preferred embodiments of DBARcontingent claim exchanges, for example in Trader and Account database261 depicted in FIG. 2, and can be displayed to traders in a formsimilar to tabulation 720 shown in FIG. 9 c, where the amount ofinvestment margin in each state is displayed for each trader, juxtaposedwith that trader's credit rating. For example, as depicted in FIG. 9 c,trader Cl who has a AAA credit rating has invested $50,000 on margin instate 7 and $100,000 on margin in state 8. In a preferred embodiment,the amount of credit risk borne by each trader can be ascertained, forexample using data from Market Data database 263 on the probability ofchanges in credit ratings (including probability of default), amountsrecoverable in case of default, correlations of credit rating changesamong the traders and the information displayed in tabulation 720.

To illustrate such determinations in the context of a group of DBARcontingent claims depicted in FIG. 9 c, the following assumptions aremade: (i) all the traders C1, C2, C3, C4 and C5 investing in the groupof contingent claims have a credit rating correlation of 0.9; (ii) theprobabilities of total default for the traders C1 to C5 are (0.001,0.003, 0.007, 0.01, 0.02) respectively; (iii) the implied probabilitiesof states S1 to S8 (depicted in FIG. 9 c) are (0.075, 0.05, 0.1, 0.25,0.2, 0.15, 0.075, 0.1), respectively. A calculation can be made withthese assumptions which approximates the total credit risk for all ofthe traders in the group of the DBAR contingent claims of FIG. 9 c,following Steps (i)-(vi) previously described for using VAR methodologyto determine Credit-Capital-at-Risk.

Step (i) involves obtaining for each trader the amount of margin used tomake each trade. For this illustration, these data are assumed and aredisplayed in FIG. 9 c, and in a preferred embodiment, are available fromTrader and Account database 261 and Trade Blotter database 266.

Step (ii) involves obtaining data related to the probability of defaultand the percentage of outstanding margin loans that are recoverable inthe event of default. In preferred embodiments, this information isavailable from such sources as the JP Morgan CreditMetrics database. Forthis illustration a recovery percentage of zero is assumed for eachtrader, so that if a trader defaults, no amount of the margin loan isrecoverable.

Step (iii) involves scaling the standard deviation of returns (in unitsof the amounts invested) by the percentage of margin used for eachinvestment, the probability of default for each trader, and thepercentage not recoverable in the event of default. For thisillustration, these steps involve computing the standard deviations ofunit returns for each state, multiplying by the margin percentage ineach state, and then multiplying this result by the probability ofdefault for each trader. In this illustration, using the assumed impliedprobabilities for states 1 through 8, the standard deviations of unitreturns are: (3.5118, 4.359,3,1.732,2,2.3805,3.5118,3). In thisillustration these unit returns are then scaled by multiplying each by(a) the amount of investment on margin in each state for each trader,and (b) the probability of default for each trader, yielding thefollowing table:

S1 S2 S3 S4 S5 S6 S7 S8 C1, 175.59 300 AAA C2, AA 285.66 263.385 C3, AA1400 999.81 C4, A+ 2598 2000 C5, A 7023.6 4359 4800

Step (iv) involves using the scaled amounts, as shown in the above tableand a correlation matrix C_(s) containing a correlation of returnsbetween each pair of defined states, in order to compute aCredit-Capital-At-Risk. As previously discussed, this Step (iv) isperformed by first arranging the scaled amounts for each trader for eachstate into a vector U as previously defined, which has dimension equalto the number of states (e.g., 8 in this example). For each trader, thecorrelation matrix C_(s) is pre-multiplied by the transpose of U andpost-multiplied by U. The square root of the result is acorrelation-adjusted CCAR value for each trader, which represents theamount of credit risk contributed by each trader. To perform thesecalculations in this illustration, the matrix C_(s) having 8 rows and 8columns and 1's along the diagonal is constructed using the methodspreviously described:

$C_{s} = \begin{matrix}1 & {- {.065}} & {- {.095}} & {- {.164}} & {- {.142}} & {- {.12}} & {- {.081}} & {- {.095}} \\{- {.065}} & 1 & {- {.076}} & {- {.132}} & {- {.115}} & {- {.096}} & {- {.065}} & {- {.076}} \\{- {.095}} & {- {.076}} & 1 & {- {.192}} & {- {.167}} & {- {.14}} & {- {.095}} & {- {.111}} \\{- {.164}} & {- {.132}} & {- {.192}} & 1 & {- {.289}} & {- {.243}} & {- {.164}} & {- {.192}} \\{- {.142}} & {- {.115}} & {- {.167}} & {- {.289}} & 1 & {- {.21}} & {- {.142}} & {- {.167}} \\{- {.12}} & {- {.096}} & {- {.14}} & {- {.243}} & {- {.21}} & 1 & {- {.12}} & {- {.14}} \\{- {.081}} & {- {.065}} & {- {.095}} & {- {.164}} & {- {.142}} & {- {.12}} & 1 & {- {.095}} \\{- {.095}} & {- {.076}} & {- {.111}} & {- {.192}} & {- {.167}} & {- {.14}} & {- {.095}} & 1\end{matrix}$

The vectors U₁, U₂, U₃, U₄, and U₅ for each of the 5 traders in thisillustration, respectively, are as follows:

$U_{1} = {{\begin{matrix}0 \\0 \\0 \\0 \\0 \\0 \\175.59 \\300\end{matrix}\mspace{14mu} U_{2}} = {{\begin{matrix}0 \\0 \\0 \\0 \\0 \\285.66 \\263.385 \\0\end{matrix}\mspace{14mu} U_{3}} = {{\begin{matrix}0 \\0 \\0 \\0 \\1400 \\999.81 \\0 \\0\end{matrix}\mspace{14mu} U_{4}} = {{\begin{matrix}0 \\0 \\0 \\2598 \\2000 \\0 \\0 \\0\end{matrix}\mspace{14mu} U_{5}} = \begin{matrix}7023.6 \\4359 \\4800 \\0 \\0 \\0 \\0 \\0\end{matrix}}}}}$

Continuing with the methodology of Step (iv) for this illustration, fivematrix computations are performed as follows:

CCAR_(i)=√{square root over (U _(i) ^(T) *C _(s) *U _(i))}

for i=1 . . . 5. The left hand side of the above equation is the creditcapital at risk corresponding to each of the five traders.

Pursuant to Step (v) of the CCAR methodology as applied to this example,the five CCAR values are arranged into a column vector of dimensionfive, as follows:

$w_{CCAR} = \begin{matrix}332.9 \\364.58 \\1540.04 \\2783.22 \\8820.77\end{matrix}$

Continuing with this step, a correlation matrix (CCAR) with a number ofrows and columns equal to the number of traders is constructed whichcontains the statistical correlation of changes in credit ratingsbetween every pair of traders on the off-diagonals and 1's along thediagonal. For the present example, the final Step (vi) involves thepre-multiplication of CCAR by the transpose of w_(CCAR) and the postmultiplication of C_(CCAR) by w_(CCAR), and taking the square root ofthat product, as follows:

CCAR_(TOTAL)=√{square root over (w _(CCAR) ^(T) *C _(CCAR) *w _(CCAR))}

In this illustration, the result of this calculation is:

$\begin{matrix}{{CCAR}_{TOTAL} = \sqrt{\begin{matrix}332.9 & 364.58 & 1540.04 & 2783.22 & 8820.77\end{matrix}*\begin{matrix}1 & {.9} & {.9} & {.9} & {.9} \\{.9} & 1 & {.9} & {.9} & {.9} \\{.9} & {.9} & 1 & {.9} & {.9} \\{.9} & {.9} & {.9} & 1 & {.9} \\{.9} & {.9} & {.9} & {.9} & 1\end{matrix}*\begin{matrix}332.9 \\364.58 \\1540.04 \\2783.22 \\8820.77\end{matrix}}} \\{= 13462.74}\end{matrix}$

In other words, in this illustration, the margin total and distributionshowing in FIG. 9 c has a single standard deviationCredit-Capital-At-Risk of $13,462.74. As described previously in thediscussion of Credit-Capital-At-Risk using VAR methodology, this amountmay be multiplied by a number derived using methods known to those ofskill in the art in order to obtain a predetermined percentile of creditloss which a trader could believe would not be exceeded with apredetermined level of statistical confidence. For example, in thisillustration, if a trader is interested in knowing, with a 95%statistical confidence, what loss amount would not be exceeded, thesingle deviation Credit-Capital-At-Risk figure of $13,462.74 would bemultiplied by 1.645, to yield a figure of $22,146.21.

A trader may also be interested in knowing how much credit risk theother traders represent among themselves. In a preferred embodiment, thepreceding steps (i)-(vi) can be performed excluding one or more of thetraders. For example, in this illustration, the most risky trader,measured by the amount of CCAR associated with it, is trader C5. Theamount of credit risk due to C1 through C4 can be determined byperforming the matrix calculation of Step (v) above, by entering 0 forthe CCAR amount of trader C5. This yields, for example, a CCAR fortraders C1 through C4 of $4,870.65.

FIG. 10 depicts a preferred embodiment of a feedback process forimproving of a system or exchange for implementing the present inventionwhich can be used with other embodiments of the present invention. Asdepicted in FIG. 10, in a preferred embodiment, closing and intraperiodreturns from Market Returns database 262 and market data from MarketData database 263 (depicted in FIG. 2) are used by process 910 for thepurpose of evaluating the efficiency and fairness of the DBAR exchange.One preferred measure of efficiency is whether a distribution of actualoutcomes corresponds to the distribution as reflected in the finalizedreturns. Distribution testing routines, such as Kolmogorov-Smirnofftests, preferably are performed in process 910 to determine whether thedistributions implied by trading activity in the form of returns acrossthe defined states for a group of DBAR contingent claims aresignificantly different from the actual distributions of outcomes forthe underlying events, experienced over

time. Additionally, in preferred embodiments, marginal returns are alsoanalyzed in process 910 in order to determine whether traders who makeinvestments late in the trading period earn returns statisticallydifferent from other traders. These “late traders,” for example, mightbe capturing informational advantages not available to early traders. Inresponse to findings from analyses in process 910, a system according tothe present invention for trading and investing in groups of the DBARcontingent claims can be modified to improve its efficiency andfairness. For example, if “late traders” earn unusually large profits,it could mean that such a system is being unfairly manipulated, perhapsin conjunction with trading in traditional security markets. Process 920depicted in FIG. 10 represents a preferred embodiment of acounter-measure which randomizes the exact time at which a tradingperiod ends for the purposes of preventing manipulation of closingreturns. For example, in a preferred embodiment, an exchange announces atrading closing end time falling randomly between 2:00 p.m. and 4:00p.m. on a given date.

As depicted in FIG. 10, process 923 is a preferred embodiment of anotherprocess to reduce risk of market manipulation. Process 923 representsthe step of changing the observation period or time for the outcome. Forexample, rather than observing the outcome at a discrete time, theexchange may specify that a range of times for observation will used,perhaps spanning many hours, day, or weeks (or any arbitrary timeframe), and then using the average of the observed outcomes to determinethe occurrence of a state.

As further depicted in FIG. 10, in response to process 910, steps couldbe taken in process 924 to modify DRFs in order, for example, toencourage traders to invest earlier in a trading period. For example, aDRF could be modified to provide somewhat increased returns to these“early” traders and proportionately decreased returns to “late” traders.Similarly for digital options, an OPF could be modified to providesomewhat discounted prices for “early” traders and proportionatelymarked-up prices for “late” traders. Such incentives, and othersapparent to those skilled in the art, could be reflected in moresophisticated DRFs.

In a preferred embodiment depicted in FIG. 10, process 921 represents,responsive to process 910, steps to change the assumptions under whichopening returns are computed for the purpose of providing better openingreturns at the opening of the trading period. For example, the resultsof process 910 might indicate that traders have excessively traded theextremes of a distribution in relation to actual outcomes. There isnothing inherently problematic about this, since trader expectations forfuture possible outcomes might reflect risk preferences that cannot beextracted or analyzed with actual data. However, as apparent to one ofskill in the art, it is possible to adjust the initial returns toprovide better estimates of the future distribution of states, by, forexample, adjusting the skew, kurtosis, or other statistical moments ofthe distribution.

As depicted in FIG. 10, process 922 illustrates changing entirely thestructure of one or more groups of DBAR contingent claims. Such acountermeasure can be used on an ad hoc basis in response to graveinefficiencies or unfair market manipulation. For example, process 922can include changes in the number of trading periods, the timing oftrading periods, the duration of a group of DBAR contingent claims, thenumber of and nature of the defined state partitions in order to achievebetter liquidity and less unfair market manipulation for groups of DBARcontingent claims of the present invention.

As discussed above (Section 6), in a preferred embodiment of a DBARDigital Options Exchange (“DBAR DOE”), traders may buy and “sell”digital options, spreads, and strips by either placing market orders orlimit orders. A market order typically is an order that isunconditional, i.e., it is executed and is viable regardless of DBARcontingent claim “prices” or implied probabilities. A limit order, bycontrast, typically is a conditional investment in a DBAR DOE in whichthe trader specifies a condition upon which the viability or execution(i.e., finality) of the order depends. In a preferred embodiment, suchconditions typically stipulate that an order is conditional upon the“price” for a given contingent claim after the trading period has beencompleted upon fulfillment of the trading period termination criteria.At this point, all of the orders are processed and a distribution ofDBAR contingent claim “prices”—which for DBAR digital options is theimplied probability that the option is “in the money”—are determined.

In a preferred embodiment of a DBAR DOE of the present invention, limitorders may be the only order type that is processed. In a preferredembodiment, limit orders are executed and are part of the equilibriumfor a group of DBAR contingent claims if their stipulated “price”conditions (i.e., probability of being in the money) are satisfied. Forexample, a trader may have placed limit buy order at 0.42 for MSFTdigital call options with a strike price of 50. With a the limitcondition at 0.42, the trader's order will be filled only if the finalDBAR contingent claim distribution results in the 50 calls having a“price” which is 0.42 or “better,” which, for a buyer of the call, means0.42 or lower.

Whether a limit order is included in the final. DBAR equilibrium affectsthe final probability distribution or “prices.” Since those “prices”determine whether such limit orders are to be executed and thereforeincluded in the final equilibrium, in a preferred embodiment aniterative procedure, as described in detail below, may be carried outuntil an equilibrium is achieved.

As described above, in a preferred embodiment, A DBAR DOE equilibriumresults for a contract, or group of DBAR contingent claims includinglimit orders, when at least the following conditions have been met:

-   -   (1) At least some buy (“sell”) orders with a limit “price”        greater (less) than or equal to the equilibrium “price” for the        given option, spread or strip are executed or “filled.”    -   (2) No buy (“sell”) orders with limit “prices” less (greater)        than the equilibrium “price” for the given option, spread or        strip are executed.    -   (3) The total amount of executed lots equals the total amount        invested across the distribution of defined states.    -   (4) The ratio of payouts should each constituent state of a        given option, spread, or strike occur is as specified by the        trader, (including equal payouts in the case of digital        options), within a tolerable degree of deviation.    -   (5) Conversion of filled limit orders to market orders for the        respective filled quantities and recalculating the equilibrium        does not materially change the equilibrium.    -   (6) Adding one or more lots to any of the filled limit orders        converted to market orders in step (5) and recalculating of the        equilibrium “prices” results in “prices” which violate the limit        “price” of the order to which the lot was added (i.e., no more        lots can be “squeaked in” without forcing market prices to go        above the limit “prices” of buy orders or below the limit        “prices” of sell orders).

In a preferred embodiment, the DBAR DOE equilibrium is computed throughthe application of limit and market order processing steps, multistatecomposite equilibrium calculation steps, steps which convert “sell”orders so that they may be processed as buy orders, and steps whichprovide for the accurate processing of limit orders in the presence oftransaction costs. The descriptions of FIGS. 11-18 which follow explainthese steps in detail. Generally speaking, in a preferred embodiment, asdescribed in Section 6, the DBAR DOE equilibrium including limit ordersis arrived at by:

-   -   (i) converting any “sell” orders to buy orders;    -   (ii) aggregating the buy orders (including the converted “sell”        orders) into groups for which the contingent claims specified in        the orders share the same range of defined states;    -   (iii) adjusting the limit orders for the effect of transaction        costs by subtracting the order fee from the order's limit        “price;”    -   (iv) sorting the orders upon the basis of the (adjusted) limit        order “prices” from best (highest) to worst (lowest);    -   (v) searching for an order with a limit “price” better (i.e.,        higher) than the market or current equilibrium “price” for the        contingent claim specified in the order;    -   (vi) if such a better order can be found, adding as many        incremental value units or “lots” of that order for inclusion        into the equilibrium calculation as possible without newly        calculated market or equilibrium “price” exceeding the specified        limit “price” of the order (this is known as the “add” step);    -   (vii) searching for an order with previously included lots which        now has a limit “price” worse than the market “price” for the        contingent claim specified in the order (i.e., lower than the        market “price”);    -   (viii) removing the smallest number of lots from the order with        the worse limit “price” so that the newly calculated equilibrium        “price,” after such iterative removal of lots, is just below the        order's limit “price” (this is known as the “prune” step, in the        sense that lots previously added are removed or “pruned” away);    -   (ix) repeating the “add” and “prune” steps until no further        orders remain which are either better than the market which have        lots to add, or worse than the market which have lots to remove;    -   (x) taking the “prices” resulting from the final equilibrium        resulting from step    -   (ix) and adding any applicable transaction fee to obtain the        offer “price” for each respective contingent claim ordered and        subtracting any applicable transaction fee to obtain the bid        “price” for each respective contingent claim ordered; and    -   (xi) upon fulfillment of all of the termination criteria related        to the event of economic significance or state of a selected        financial product, allocating payouts to those orders which have        investments on the realized state, where such payouts are        responsive to the final equilibrium “prices” of the orders'        contingent claims and the transaction fees for such orders.

Referring to FIG. 11, illustrative data structures are depicted whichmay be used in a preferred embodiment to store and manipulate the datarelevant to the DBAR DOE embodiment and other embodiments of the presentinvention. The data structure for a “contract” or group of DBARcontingent claims, shown in 1101, contains data members which store datawhich are relevant to the construction of the DBAR DOE contract or groupof claims. Specifically, the contract data structure contains (i) thenumber of defined states (contract.numStates); (ii) the total amountinvested in the contract at any given time (contract.totalInvested);(iii) the aggregate profile trade investments required to satisfy theaggregate profile trade requests for profile trades (a type of tradewhich is described in detail below) (iv) the aggregate payout requestsmade by profile trades; (v) the total amount invested or allocated ineach defined state at any given time (contract.stateTotal); (vi) thenumber of orders submitted at any given time (contract.numOrders); and(vii) a list of the orders, which is itself a structure containing datarelevant to the orders (contract.orders[ ]).

A preferred embodiment of “order” data structures, shown in 1102 of FIG.11, illustrates the data which are typically needed to process atrader's order using the methods of the DBAR DOE of the presentinvention. Specifically, the order data structure contains the followingrelevant members for order processing:

the amount of the order which the trader desires to transact. For orderswhich request the purchase (“buys”) of a digital option, strip, orspread, the amount is interpreted as the amount to invest in the desiredcontingent claim. Thus, for buys, the order amount is analogous to theoption premium for conventional options. For orders which request“sales” of a DBAR contingent claim, the order amount is to beinterpreted as the amount of net payout that the trader desires to“sell.” Selling a net payout in the context of a DBAR DOE of the presentinvention means that the loss that a trader suffers should the digitaloption, strip or spread “sold” expire in the money is equal to thepayout “sold.” In other words, by selling a net payout, the trader isable to specify the amount of net loss that would occur should theoption “sold” expire in the money. If the contingent claim “sold”expires out of the money, the trader would receive a profit equal to thenet payout multiplied by the ratio of (a) the final implied probabilityof the option expiring in the money and (b) the implied probability ofthe option expiring out of the money. In other words, in a preferredembodiment of a DBAR DOE, “buys are for premium, and sells are for netpayout” which means that buy orders and sell orders in terms of theorder amount are interpreted somewhat differently. For a buy order, thepremium is specified and the payout, should the option expire in themoney, is not known until all of the predetermined termination criteriahave been met at the end of trading. For a “sell” order, in contrast,the payout to be “sold” is specified (and is equal to the net lossshould the option “sold” expire in the money), while the premium, whichis equal to the trader's profit should the option “sold” expire out ofthe money, is not known until all of the predetermined terminationcriteria have been met (e.g., at the end of trading);the amount which must be invested in each defined state to generate thedesired digital option, spread or strip specified in the order iscontained in data member order.invest[ ]; the data members order.buySellindicates whether the order is a buy or a “sell”; the data membersorder.marketLimit indicates whether the order is a limit order whoseviability for execution is conditional upon the final equilibrium“price” after all predetermined termination criteria have been met, or amarket order, which is unconditional;the current equilibrium “price” of the digital option, spread or stripspecified in the order; a vector which specifies the type of contingentclaim to be traded (order.ratio[ ]). For example, in a preferredembodiment involving a contract with seven defined states, an order fora digital call option which would expire in the money should any of thelast four states occur would be rendered in the data member order.ratio[] as order.ratio[0,0,0,1,1,1,1] where the 1's indicate that the samepayout should be generated by the multistate allocation process when thedigital option is in the money, and the 0's indicate that the option isout of the money, or expires on one of the respective out of the moneystates. As another example in a preferred embodiment, a spread which isin the money should states either states 1,2, 6, or 7 occur would berendered as order.ratio[1,1,0,0,0,1,1]. As another example in apreferred embodiment, a digital option strip, which allows a trader tospecify the relative ratios of the final payouts owing to an investmentin such a contingent claim would be rendered using the ratios over whichthe strip is in the money. For example, if a trader desires a stripwhich pays out three times much as state 3 should state 1 occur, andtwice as much as state 3 if state 2 occurs, the strip would be renderedas order.ratio[3,2,1,0,0,0,0];the amount of the order than can be executed or filled at equilibrium.For market orders, the entire order amount will be filled, since suchorders are unconditional. For limit orders, none, all, or part of theorder amount may be filled depending upon the equilibrium “prices”prevailing when the termination criteria are fulfilled;the transaction fee applicable to the order;the payout for the order, net of fees, after all predeterminedtermination criteria have been met; anda data structure which, for trades of the profile type (described belowin detail), contains the desired amount of payout requested by the ordershould each state occur.

FIG. 12 depicts a logical diagram of the basic steps for limit andmarket order processing in a preferred embodiment of a DBAR DOE of thepresent invention. Step 1201 of FIG. 12 loads the relevant data into thecontract and order data structures of FIG. 11. Step 1202 initializes theset of DBAR contingent claims, or the “contract,” by placing initialamounts of value units (i.e., initial liquidity) in each state of theset of defined states. The placement of initial liquidity avoids asingularity in any of the defined states (e.g., an invested amount in agiven defined state equal to zero) which may tend to impede multistateallocation calculations. The initialization of step 1202 may be done ina variety of different ways. In a preferred embodiment, a small quantityof value units is placed in each of the defined states. For example, asingle value unit (“lot”) may be placed in each defined state where thesingle value unit is expected to be small in relation to the totalamount of volume to be transacted. In step 1202 of FIG. 12, the initialvalue units are represented in the vector init[contract.numStates].

In a preferred embodiment, step 1203 of FIG. 12 invokes the functionconvertSales( ) which converts all of the “sell” orders to complementarybuy orders. The function convertSales( ) is described in detail in FIG.15, below. After the completion of step 1203, all of the orders forcontingent claims—whether buy or “sell” orders, can be processed as buyorders.

In a preferred embodiment, step 1204 groups these buy orders based uponthe distinct ranges of states spanned by the contingent claims specifiedin the orders. The range of states comprising the order are contained inthe data member order.ratio[ ] of the order data structure 1102 depictedin FIG. 11.

In a preferred embodiment, for each order[j] there is associated avector of length equal to the number of defined states in the contractor group of DBAR contingent claims (contract.numStates). This vector,which is stored in order[j].ratio[ ], contains integers which indicatethe range of states in which an investment is to be made in order togenerate the expected payout profile of the contingent claim desired bythe trader placing the order.

In a preferred embodiment depicted in FIG. 12, a separate grouping instep 1204 is required for each distinct order[j].ratio[ ] vector. Twoorder[j].ratio[ ] vectors are distinct for different orders when theirdifference yields a vector that does not contain zero in every element.For example, for a contract which contains seven defined states, adigital put option which spans that first three states has anorder[1].ratio[ ] vector equal to (1,1,1,0,0,0,0). A digital call optionwhich spans the last five states has an order[2].ratio[ ] vector equalto (0,0,1,1,1,1,1). Because the difference of these two vectors is equalto (1,1,0,−1,−1,−1,−1), these two orders should be placed into distinctgroups, as indicated in step 1204.

In a preferred embodiment depicted in FIG. 12, step 1204 aggregatesorders into relevant groups for processing. For the purposes ofprocessing limit orders: (i) all orders may be treated as limit orderssince orders without limit “price” conditions, e.g., “market orders,”can be rendered as limit buy orders (including “sale” orders convertedto buy orders in step 1203) with limit “prices” of 1, and (ii) all ordersizes are processed by treating them as multiple orders of the smallestvalue unit or “lot.”

The relevant groups of step 1204 of FIG. 12 are termed “composite” sincethey may span, or comprise, more than one of the defined states. Forexample, the MSFT Digital Option contract depicted above in Table 6.2.1,for example, has defined states (0,30], (30,40], (40,50], (50,60], (60,70], (70, 80], and (80,00]. The 40 strike call options therefore spanthe five states (40,50], (50,60], (60, 70], (70, 80], and (80,00]. A“sale” of a 40 strike put, for example, would be converted at step 1203into a complementary buy of a 40 strike call (with a limit “price” equalto one minus the limit “price” of the sold put), so both the “sale” ofthe 40 strike put and the buy of a 40 strike call would be aggregatedinto the same group for the purposes of step 1204 of FIG. 12.

In the preferred embodiment depicted in FIG. 12, step 1205 invokes thefunction feeAdjustOrders( ) This function is required so as toincorporate the effect of transaction or exchange fees for limit orders.The function feeAdjustOrders( ) shown in FIG. 12, described in detailwith reference to FIG. 16, basically subtracts from the limit “price” ofeach order the fee for that order's contingent claim. The limit “price”is then set to this adjusted, lower limit “price” for the purposes ofthe ensuing equilibrium calculations. At the point of step 1206 of thepreferred embodiment depicted in FIG. 12, all of the orders may beprocessed as buy orders (because any “sell” orders have been convertedto buy orders in step 1203 of FIG. 12) and all limit “prices” have beenadjusted (with the exception of market orders which, in a preferredembodiment of the DBAR DOE of the present invention, have a limit“price” equal to one) to reflect transaction costs equal to the feespecified for the order's contingent claim (as contained in the datamember order[j].fee). For example, consider the steps depicted in FIG.12 leading up to step 1206 on three hypothetical orders: (1) a buy orderfor a digital call with strike price of 50 with a limit “price” of 0.42for 100,000 value units (lots) (on the illustrative MSFT exampledescribed above); (2) a “sale” order for a digital put with a strikeprice of 40 with a limit price of 0.26 for 200,000 value units (lots);and (3) a market buy order for a digital spread which is in the moneyshould MSFT stock expire greater than or equal to 40 and less than orequal to 70. In a preferred embodiment, the representations of the rangeof states for the contingent claims specified in the three orders are asfollows: (1) buy order for 50-strike digital call: order[1].ratio[]=(0,0,0,1,1,1,1); (2) “sell” order for 40-strike digital put:order[2].ratio[ ]=(0,0,1,1,1,1,1); and (3) market buy order for adigital spread in the money on the interval [40,70): order[3].ratio[]=(0,0,1,1,1,1,0). Also in this preferred embodiment, the “sell” orderof the put covers the states as a “converted” buy order which arecomplementary to the states being sold (sold states=order.ratio[]=(1,1,0,0,0,0,0)), and the limit “price” of the converted order isequal to one minus the limit “price” of the original order (i.e.,1−0.26=0.74). Then in a preferred embodiment, all of the orders' limit“prices” are adjusted for the effect of transaction fees so that,assuming a fee for all of the orders equal to 0.0005 (i.e., 5 basispoints of notional payout), the fee-adjusted limit prices of the ordersare equal to (1) for the 50-strike call: 0.4195 (0.42−0.0005); (2) forthe converted sale of 40-strike put: 0.7395 (1−0.26−0.0005); and (3) forthe market order for digital spread: 1 (limit “price” is set to unity).In a preferred embodiment depicted in FIG. 12, step 1204 then wouldaggregate these hypothetical orders into distinct groups, where ordersin each group share the same range of defined states which comprise theorders' contingent claim. In other words, as a result of step 1204, eachgroup contains orders which have identical vectors in order.ratio[ ].For the illustrative three hypothetical orders, the orders would beplaced as a result of step 1204 into three separate groups, since eachorder ranges over distinct sets of defined states as indicated in theirrespective order[j].ratio[ ] vectors (i.e., (0,0,0,1,1,1,1),(0,0,1,1,1,1,1), and (0,0,1,1,1,1,0), respectively).

For the purposes of step 1206 of the preferred embodiment depicted inFIG. 12, all of the order have been converted to buy orders and have hadtheir limit “prices” adjusted to reflect transaction fees, if any. Inaddition, such orders have been placed into groups which share the samerange of defined states which comprise the contingent claim specified inthe orders (i.e., have the same order[j].ratio[ ] vector). In thispreferred embodiment depicted in FIG. 12, step 1206 sorts each group'sorders based upon their fee-adjusted limit “prices,” from best (highest“prices”) to worst (lowest “prices”). For example, consider a set oforders in which only digital calls and puts have been ordered, both tobuy and to “sell,” for the MSFT example of Table 6.2.1 in which strikeprices of 30, 40, 50, 60, 70, and 80 are available. A “sale” of a callis converted to a buy of a put, and a “sale” of a put is converted intoa purchase of a call by step 1204 of the preferred embodiment depictedin FIG. 12. Thus, in this embodiment all of the grouped orderspreferably are grouped in terms of calls and puts at the indicatedstrike prices of the orders.

The grouped orders, after conversion and adjustment for fees, can beillustrated in the following Diagram 1, which depicts the results of agrouping process for a set of illustrative and assumed digital puts andcalls.

Referring to Diagram 1 the call and put limit orders have been groupedby strike price (distinct order[j].ratio[ ] vectors) and then orderedfrom “best price” to “worst,” moving away from the horizontal axis. Asshown in the table, “best price” for buy orders are those with higherprices (i.e., buyers with a higher willingness to pay). Diagram 1includes “sales” of puts which have been converted to complementarypurchases of calls and “sales” of calls which have been converted tocomplementary purchases of puts, i.e., all orders for the purposes ofDiagram 1 may be treated as buy orders.

For example, as depicted in Diagram 1 the grouping which includes thepurchase of the 40 calls (labeled “C40”) would also include anyconverted “sales” of the 40 puts (i.e., “sale” of the 40 puts has anorder.ratio[ ] vector which originally is equal to (1,1,0,0,0,0,0) andis then converted to the complementary order.ratio[ ] vector(0,0,1,1,1,1,1) which corresponds to the purchase of a 40-strike call).

Diagram 1 illustrates the groupings that span distinct sets of definedstates with a vertical bar. The labels within each vertical bar inDiagram 1 such as “C50”, indicate whether the grouping corresponds to acall (“C”) or put (“P”) and the relevant strike price, e.g., “C50”indicates a digital call option with strike price of 50.

The horizontal lines within each vertical bar shown on Diagram 1indicates the sorting by price within each group. Thus, for the verticalbar above the horizontal axis marked “C50” in Diagram 1, there are fivedistinct rectangular groupings within the vertical bar. Each of thesegroupings is an order for the digital call options with strike price 50at a particular limit “price.” By using the DBAR methods of the presentinvention, there is no matching of buyers and “sellers,” or buy ordersand “sell” orders, which is typically required in the traditionalmarkets in order for transactions to take place. For example, Diagram 1illustrates a set of orders that contains only buy orders for thedigital puts struck at 70 (“P70”).

In a preferred embodiment of a DBAR DOE of the present invention, theaggregation of orders into groups referred to by step 1204 of thepreferred embodiment depicted in FIG. 12 corresponds to DBAR digitaloptions, spread, and strip trades which span distinct ranges of thedefined states. For example, the 40 puts and the 40 calls arerepresented as distinct state sets since they span or comprise differentranges of defined states.

Proceeding with the next step of the preferred embodiment depicted inFIG. 12, step 1207 queries whether there is at least a single orderwhich has a limit “price” which is “better” than the current equilibrium“price” for the ordered option. In a preferred embodiment for the firstiteration of step 1207 for a trading period for a group of DBARcontingent claims, the current equilibrium “prices” reflect theplacement of the initial liquidity from step 1202. For example, with theseven defined states of the MSFT example described above, one value unitmay have been initialized in each of the seven defined states. The“prices” of the 30, 40, 50, 60, 70, and 80 digital call options, aretherefore 6/7, 5/7, 4/7, 3/7, 2/7, and 1/7, respectively. The initial“prices” of the 30, 40, 50, 60, 70, and 80 digital puts are 1/7, 2/7,3/7, 4/7, 5/7, 6/7, respectively. Thus, step 1207 may identify a buyorder for a 60 digital call option with limit “price” greater than 3/7(0.42857) or a “sell” order, for example, for the 40 digital put optionwith limit “price” less than 2/7 (0.28571) (which would be convertedinto a buy order of the 40 calls with limit “price” of 5/7 (i.e.,1−2/7)). In a preferred embodiment an order's limit “price” or impliedprobability would take into account transaction or exchange fees, sincethe limit “prices” of the original orders would have been alreadyadjusted by the amount of the transaction fee (as contained inorder[j].fee) from step 1205 of FIG. 12, where the function fee AdjustOrders( ) is invoked. As discussed above, transaction or exchange fees,and consequently bid/offer “prices” or implied probability, can becomputed in a variety of ways. In a preferred embodiment, such fees arecomputed as a fixed percentage of the total amount invested over all ofthe defined states. The offer (bid) side of the market for a givendigital option (or strip or spread) is computed in this embodiment bytaking the total amount invested less (plus) this fixed percentage, anddividing it by the total amount invested over the range of statescomprising the given option (or strip or spread). This reciprocal ofthis quantity then equals the offer (bid) “price” in this embodiment. Inanother preferred embodiment, transaction fees are computed as a fixedpercentage of the payout of a given digital option, strip or spread. Inthis embodiment, if the transaction fee is f basis points of the payout,then the offer (bid) price is equal to the total amount invested overthe range of state comprising the digital option (strip or spread) plus(minus) f basis points. For example, assume that f is equal to 5 basispoints or 0.0005. Thus, the offer “price” of an in-the-money optionwhose equilibrium “price” is 0.50 might be equal to 0.50+0.0005 or0.5005 and the bid “price” equal to 0.50−0.0005 or 0.4995. Anout-of-the-money option having an equilibrium “price” equal to 0.05might therefore have an offer “price” equal to 0.05+0.0005 or 0.0505 anda bid “price” equal to 0.05−0.0005 or 0.0495. Thus, the embodiment inwhich transaction fees are a fixed percentage of the payout yieldsbid/offer spreads that are a higher percentage of the out-of-the-moneyoption “prices” than of the in-the-money option prices.

The bid/offer “prices” affect not only the costs to the trader of usinga DBAR digital options exchange, but also the nature of the limit orderprocess. Buy limit orders (including those buy orders which areconverted “sells”) must be compared to the offer “prices” for theoption, strip or spread contained in the order. Thus a buy order has alimit “price” which is “better” than the market if the limit “price”condition is greater than or equal to the offer side of the market forthe option specified in the order. Conversely, a “sell” order has alimit “price” which is better than the market if the limit “price”condition is less than or equal to the bid side of the market for theoption specified in the order. In the preferred embodiment depicted inFIG. 12, the effect of transaction fees is captured by the adjustment ofthe limit “prices” in step 1205, in that in equilibrium an order shouldbe filled only if its limit “price” is better than the offer “price”,which includes the transaction fee.

In the preferred embodiment depicted in FIG. 12, if step 1207 identifiesat least one order which has a limit “price” better than the current setof equilibrium “prices” (whether the initial set of “prices” upon thefirst iteration or the “prices” resulting from subsequent iterations)then step 1208 invokes the function fillRemoveLots. The functionfillRemoveLots, when called with the first parameter equal to one as instep 1208, will attempt to add lots from the order identified in step1207 which has limit “price” better than the current set of equilibriumprices. The fillRemoveLots function is described in detail in FIG. 17,below. Basically, the function finds the number of lots of the orderthan can be added for a buy order (including all “sale” order convertedto buy orders) such that when a new equilibrium set of “prices” iscalculated for the group of DBAR contingent claims with the added lots(by invoking the function compEq( ) of FIG. 13), no further lots can beadded without causing the new equilibrium “price” with those added lotsto exceed the limit “price” of the buy order being filled.

In preferred embodiments, finding the maximum amount of lots to add sothat the limit “price” is just better than the new equilibrium isaccomplished using the method of binary search, as described in detailwith reference to FIG. 17, below. Also in preferred embodiments the stepof “filling” lots refers to the execution, incrementally anditeratively, using the method of binary search, of that part of theorder quantity that can be executed or “filled.” In a preferredembodiment, the filling of a buy order therefore requires the testing,via the method of binary search, to determine whether additional unitlots can be added over the relevant range of defined states spanning theparticular option for the purposes of equilibrium calculation, withoutcausing the resulting equilibrium “price” for the order to exceed thelimit “price.”

In the preferred embodiment depicted in FIG. 12, step 1209 is executedfollowing step 1208 if lots are filled, or following step 1207 if noorders were identified with limit “prices” which are better than thecurrent equilibrium “prices.” Step 1209 of FIG. 12 identifies ordersfilled at least partially at limit “prices” which are worse (i.e., less)than the current equilibrium “prices.” In preferred embodiments, thefilling of lots in step 1208, if performed prior to step 1209, involvesthe iterative recalculation of the equilibrium “prices” by invoking thefunction compEq( ) which is described in detail with reference to FIG.13.

In the preferred embodiment depicted in FIG. 12, the equilibriumcomputations in step 1208 performed in the process of filling lots maycause a change in the equilibrium “prices” which in turn may causepreviously filled orders to have limit “prices” which are now worse(i.e., lower) than the new equilibrium. Step 1209 identifies theseorders. In order for the order to comply with the equilibrium, its limit“price” may not be worse (i.e., less) than the current equilibrium.Thus, in a preferred embodiment of the DBAR DOE of the presentinvention, lots for such an order are removed. This is performed in step1210 with the invocation of function fillRemoveLots. Similar to step1208, in a preferred embodiment the processing step 1210 uses the methodof binary search to find the minimum amount of lots to be removed fromthe quantity of the order that has already been filled such that theorder's limit “price” is no longer worse (i.e., less) than theequilibrium “price,” which is recomputed iteratively. For buy orders andall buy orders converted from “sell” orders processed in step 1210, anew filled quantity is found which is smaller than the original filledquantity so that the buy order's new equilibrium “price” does not exceedthe buy order's specified limit “price.”

The logic of steps 1207-1210 of FIG. 12 may be summarized as follows. Anorder is identified which can be filled (step 1207), i.e., an orderwhich has a limit “price” better than the current equilibrium “price”for the option specified in the order. If such an order is identified,it is filled to the maximum extent possible without violating the limit“price” condition of the order itself (step 1208). A buy order's limit“price” condition is violated if an incremental lot is filled whichcauses the equilibrium “price,” taking account of this additional lot,to exceed the buy order's limit “price.” Any previously filled ordersmay now have limit order conditions that are violated as a result oflots being filled in step 1208. These orders are identified, one orderat a time, in step 1209. The filled amounts of such orders with violatedlimit order “price” conditions are reduced or “pruned” so that the limitorder “price” conditions are no longer violated. This “pruning” isperformed in step 1210. The steps 1207 to 1210 constitute an “add andprune” cycle in which an order with a limit “price” better than theequilibrium of the current iteration has its filled amount increased,followed by the reduction or pruning of any filled amounts for orderswith a limit “price” condition which is worse than the equilibrium“price” of the current iteration.

In preferred embodiment, the “add and prune” cycle continues until thereremain no further orders with limit “price” conditions which are eitherbetter or worse than the equilibrium, i.e., no further adding or pruningcan be performed.

When no further adding or pruning can be performed, an equilibrium hasbeen achieved, i.e., all of the orders with limit “prices” worse thanthe equilibrium are not executed and at least some part of all of theorders with limit “prices” better or equal to the equilibrium areexecuted. In the preferred embodiment of FIG. 12, completion of the “addand prune” cycle terminates limit and market order processing asindicated in step 1211. The final “prices” of the equilibriumcalculation resulting from the “add and prune” cycle of steps 1207-1210can be designated as the mid-market “prices.” The bid “prices” for eachcontingent claim are computed by subtracting a fee from the mid-market“prices,” and the offer “prices” are computed by adding a fee to themid-market “prices.” Thus, equilibrium mid-market, bid, and offer“prices” may then be published to traders in a preferred embodiment of aDBAR DOE.

Referring now to the preferred embodiment of a method of compositemultistate equilibrium calculation depicted FIG. 13, the functioncompEq( ) which is a multistate allocation algorithm, is described. In apreferred embodiment of a DBAR DOE, digital options span or comprisemore than one defined state, with each of the defined statescorresponding to at least one possible outcome of an event of economicsignificance or a financial instrument. As depicted in Table 6.2.1above, for example, the MSFT digital call option with strike price of 40spans the five states above 40 or (40,50], (50,60], (60, 70], (70, 80],and (80,00]. To achieve a profit and loss scenario that tradersconventionally expect from a digital option, in a preferred embodimentof the present invention a digital option investment of value unitsdesignates a set of defined states and a desired return-on-investmentfrom the designated set of defined states, and the allocation ofinvestments across these states is responsive to the desiredreturn-on-investment from the designated set of defined states. For adigital option, the desired return on investment is often expressed as adesire to receive the same payout regardless of the state that occursamong the set of defined states that comprise the digital option. Forinstance, in the illustrative example of the MSFT stock prices shown inFIG. 6.2.1, a digital call option with strike price of 40 would be, in apreferred embodiment, allocated the same payout irrespective of whichstate of the five states above 40 occurs.

In preferred embodiments of the DBAR DOE of the present invention,traders who invest in digital call options (or strips or spreads)specify a total amount of investment to be made (if the amount is for abuy order) or notional payout to be “sold” (if the amount is for a“sell” order). In a preferred embodiment, the total investment is thenallocated using the compEq( ) multistate allocation method depicted inFIG. 13. In another preferred embodiment, the total amount of the payoutto be received, should the digital option expire in the money, isspecified by the investor, and in a preferred embodiment the investmentamount required to produce such payouts are computed by the multistateallocation method depicted in FIG. 14.

In either embodiment, the investor specifies a desired return oninvestment from a designated set of defined states. A return oninvestment is the amount of value units received from the investmentless the amount of value units invested, divided by the amount invested.In the embodiment depicted in FIG. 13, the amount of value unitsinvested is specified and the amount of value units received, or thepayout from the investment, is unknown until the termination criteriaare fulfilled and the payouts are calculated. In the embodiment depictedin FIG. 14, the amount of value units to be paid out is specified butthe investment amount to achieve that payout it is unknown until thetermination criteria are fulfilled. The embodiment depicted in FIG. 13is known, for example, as a composite trade, and the embodiment depictedin FIG. 14 is known, for example, as a profile trade.

Referring back to FIG. 13, step 1301 invokes a function call to thefunction profEq( ) This function handles those types of trades in whicha desired return-on-investment for a designated set of defined states isspecified by the trader indicating the payout amount to be receivedshould any of the designated set of defined states occur. For example, atrader may indicate that a payout of $10,000 should be received shouldthe MSFT digital calls struck at 40 finish in the money. Thus, if MSFTstock is observed at the expiration date to have a price of 45, theinvestor receives $10,000. If the stock price were to be below 40, theinvestor would lose the amount invested, which is calculated using thefunction profEq( ) This type of desired return-on-investment trade isreferred to as a multistate profile trade, and FIG. 14 depicts thedetailed logical steps for a preferred embodiment of the profEq( )function. In preferred embodiments of a DBAR DOE of the presentinvention, there need not be any profile trades.

Referring back to FIG. 13, step 1302 initializes control loop countervariables. Step 1303 indicates a control loop that executes for eachorder. Step 1304 initializes the variable “norm” to zero and assigns theorder being processed, order[j], to the order data structure. Step 1305begins a control loop that executes for each of defined states thatcomprise a given order. For example, the MSFT digital call option withstrike of 40 illustrated in Table 6.2.1 spans the five states that rangefrom 40 and higher.

In the preferred embodiment depicted in FIG. 13, step 1306 executeswhile the number of states in the order are being processed to calculateof the variable norm, which is the weighted sum of the total investmentsfor each state of the range of defined states which comprise the order.The weights are contained in order.ratio[i], which is a vector typemember of the order data structure illustrated in FIG. 11 as previouslydescribed. For digital call options, whose payout is the same regardlessof the defined state which occurs over the range of states for which thedigital option is in the money, all of the elements of order.ratio[ ]are equal over the range. For trades involving digital strips, theratios in order.ratio[ ] need not be equal. For example, a trader maydesire a payout which is twice as great should a range of states occurcompared to another range of states. The data member order.ratio[ ]would therefore contain information about this desired payout ratio.

In the preferred embodiment depicted in FIG. 13, after all of the statesin the range of states spanning the order have been processed, thecontrol loop counter variable is re-initialized in step 1307, step 1308begins another control loop the defined states spanning the order. Inpreferred embodiments, step 1309 calculates the amount of the investmentspecified by the order that must be invested in each defined statespanning the range of states for the order. Sub-step 2 of step 1309contains the allocation which is assigned to order.invest[i], for eachof these states. This sub-step allocates the amount to be invested in anin-the-money state in proportion to the existing total investment inthat state divided by the sum of all of the investment in thein-the-money states. Sub-steps 3 and 4 of step 1309 add this allocationto the investment totals for each state (contract.stateTotal[state]) andfor all of the states (contract.totallnvested) after subtracting out theallocation from the previous iteration (temp). In this manner, theallocation steps proceed iteratively until a tolerable level of errorconvergence is achieved.

After all of the states in the order have been allocated in 1309, step1310 of the preferred embodiment depicted in FIG. 13 calculates the“price” or implied probability of the order. The “price” of the order isequal to the vector product of the order ratio (a vector quantitycontained in order.ratio[ ]) and the total invested in each state (avector quantity contained in contract.stateTotal[ ]) divided by thetotal amount invested over all of the defined states (contained incontract.totallnvested), after normalization by the maximum value in thevector order.ratio[ ]. As further depicted in step 1310 the resulting“price” for the digital option, strip, or spread is stored in the pricemember of the order data structure (order.price).

In the preferred embodiment of the method of multistate compositeequilibrium calculation for a DBAR DOE of the present invention. Step1311 moves the order processing step to the next order. After all of theorders have been processed, step 1312 of the preferred embodimentdepicted in FIG. 13 calculates the level of error, which is based uponthe percentage deviations of the payouts resulting from the previousiteration to the payouts expected by the trader. If the error istolerably low (e.g., epsilon=10⁴), the compEq( ) function terminates(step 1314). If the error is not tolerably low, then compEq( ) isiterated again, as shown in step 1313.

FIG. 14 depicts a preferred embodiment of a method of multistate profileequilibrium calculation in a DBAR DOE of the present invention. As shownin FIG. 14, when a new multistate profile trade is added, the functionaddProfile( ) of step 1401 adds information about the trade to the datastructure members of the contract data structure, as described above inFIG. 11. The first step of the profEq( ) function, step 1402, shows thatthe profEq( ) function proceeds iteratively until a tolerable level ofconvergence is achieved, i.e., an error below some error parameterepsilon (e.g., 10⁻⁸). If the error objective has not been met, in apreferred embodiment all of the previous allocations from any priorinvocations of profEq( ) are subtracted from the total investments ineach state and from the total investment for all of the states, asindicated, in step 1405. This is done for each of the states, asindicated in control loop 1404 after initialization of the loop counter(step 1403).

In a preferred embodiment, the next step, step 1406, computes theinvestment amount necessary to generate the desired return-on-investmentwith a fixed payout profile. Sub-step 1 of 1406 shows that theinvestment amount required to achieve this payout profile for a state isa positive solution to the quadratic equation CDRF 3 set forth inSection 2.4 above. In the preferred embodiment depicted in FIG. 14, thesolution, contract.poTrade[i], is then added to the total investmentamount in that state as indicated in sub-step 2 of step 1406. The totalinvestment amount for all of the states is also increased bycontract.poTrade[i], and sub-step 4 of 1406 increments the control loopcounter for the number of states. In the preferred embodiment depictedin FIG. 14, the calculation of the quadratic equation of sub-step 3 ofstep 1406 is completed for each of the states, and then repeatediteratively until a tolerable level of error is achieved.

FIG. 15 depicts a preferred embodiment of a method for converting “sell”orders to buy orders in a DBAR DOE of the present invention. The methodis contained in the function convertSales( ) called within the limit andmarket order processing steps previously discussed with reference toFIG. 11.

As discussed above in a preferred embodiment of a DBAR DOE, buy ordersand “sell” order are interpreted somewhat differently. The amount of abuy order (as contained in the data structure member order.orderAmount)is interpreted as the amount of the investment to be allocated over therange of states spanning the contingent claim specified in the order.For example, a buy order for 100,000 value units for an MSFT digitalcall with strike price of 60 (order.ratio[ ]=(0,0,0,0,1,1,1) in the MSFTstock example depicted in Table 6.2.1) will be allocated among thestates comprising the order so that, in the case of a digital option,the same payout is received regardless of which constituent state of therange of states is realized. For a “sell” order the order amount (asalso contained in the member data structure order.orderAmount) isinterpreted to be the amount which the trader making the sale stands tolose if the contingent claim (i.e., digital option, spread, or strip)being “sold” expires in the money (i.e., any of the constituent statescomprising the sale order is realized). Thus, the “sale” order amount isinterpreted as a payout (or “notional” or “notional payout”) less theoption premium “sold,” which is the amount that may be lost should thecontingent claim “sold” expire in the money (assuming, that is, theentire order amount can be executed if the order is a limit order). Abuy order, by contrast, has an order amount which is interpreted as aninvestment amount which will generate a payout whose magnitude is knownonly after the termination of trading and the final equilibrium pricesfinalized, should the option expire in the money. Thus, a buy order hasa trade amount which is interpreted as in investment amount or option“premium” (using the language of the conventional options markets)whereas a DBAR DOE “sell” order has an order amount which is interpretedto be a net payout equal to the gross payout lost, should the optionsold expire in the money, less the premium received from the “sale.”Thus, in a preferred embodiment of a DBAR DOE, buy orders have orderamounts corresponding to premium amounts, while “sell” orders have orderamounts corresponding to net payouts.

One advantage of interpreting the order amount of the buy and “sell”orders differently is to facilitate the subsequent “sale” of a buy orderwhich has been executed (in all or part) in a previous trading period.In the case where a subsequent trading period on the same underlyingevent of economic significance or state of a financial product isavailable, a “sale” may be made of a previously executed buy order froma previous and already terminated and finalized trading period, eventhough the observation period may not be over so that it is not knownwhether the option finished in the money. The previously executed buyorder, from the earlier and finalized trading period, has a known payoutamount, should the option expire in the money. This payout amount isknown since the earlier trading period has ended and the finalequilibrium “prices” have been calculated. Once a subsequent tradingperiod on the same underlying event of economic significance is open fortrading (if such a trading period is made available), a trader who hasexecuted the buy order may then sell it by entering a “sell” order withan order. The amount of the “sell” order can be a function of thefinalized payout amount of the buy order (which is now known withcertainty, should the previously bought contingent claim expire in themoney), and the current market price of the contingent claim being“sold.” Setting this order amount of the “sale” equal to y, the tradermay enter a “sale” such that y is equal to:

y=P*(1−g)

where P is the known payout from the previously finalized buy order froma preceding trading period, and q is the “price” of the contingent claimbeing “sold” during the subsequent trading period. In preferredembodiments, the “seller” of the contingent claim in the second periodmay enter in a “sale” order with order amount equal to y and a limit“price” equal to q. In this manner the trader is assured of “selling”his claim at a “price” no worse than the limit “price” equal to q.

Turning now to the preferred embodiment of a method for converting“sale” ordeth to buy orders depicted in FIG. 15, in step 1501 a controlloop is initiated of orders (contract.numOrders). Step 1502 querieswhether the order under consideration in the loop is a buy(order.buySell=1) or a “sell” order (order.buySell=−1). If the order isa buy order then no conversion is necessary, and the loop is incrementedto the next order as indicated in step 1507.

If, on the other hand, the order is a “sell” order, then in preferredembodiments of the DBAR DOE of the present invention conversion isnecessary. First, the range of states comprising the contingent claimmust be changed to the complement range of states, since a “sale” of agiven range of states is treated as equivalent to a buy order for thecomplementary range of states. In the preferred embodiment of FIG. 15,step 1503 initiates a control loop to execute for each of the definedstates in the contract (contract.numStates), step 1504 does theswitching of the range of states sold to the complementary states to bebought. This is achieved by overwriting the original range of statescontained in order[j].ratio[ ] to a complement range of states. In thispreferred embodiment, the complement is equal to the maximum entry forany state in the original order[j].ratio[ ] vector (for each order)minus the entry for each state in order[j].ratio[ ]. For example, if atrader has entered an order to sell 50-strike puts in MSFT exampledepicted in Table 6.2.1, then originally order.ratio[ ] is the vector(1,1,1,0,0,0,0), i.e., 1's are entered which span the states (0,30],(30,40], (40,50] and zeroes are entered elsewhere. To obtain thecomplement states to be bought, the maximum entry in the originalorder.ratio[ ] vector for the order is obtained. For the put option tobe “sold,” the maximum of (1,1,1,0,0,0,0) is clearly 1. Each element ofthe original order.ratio[ ] vector is then subtracted from the maximumto produce the complementary states to be bought. For this example, theresult of this calculation is (0,0,0,1,1,1,1), i.e., a purchase of a50-strike call is complementary to the “sale” of the 50-strike put. Iffor example, the original order was for a strip in which the entries inorder.ratio[ ] are not equal, in a preferred embodiment the samecalculation method would be applied. For example, a trader may desire to“sell” a payout should any of the same three states which span the50-strike put occur, but desires to sell a payout of three times theamount of state (40,50] should state (0,30] occur and sell twice thepayout of (40,50] should state (30,40] occur. In this example, theoriginal order.ratio for the “sale” of a strip is equal to(3,2,1,0,0,0,0). The maximum value for any state of this vector is equalto 3. The complementary buy vector is then equal to each element of theoriginal vector subtracted from the maximum, or (0, 1, 2, 3,3,3,3).Thus, a “sale” of the strip (3,2,1,0,0,0,0) is revised to a purchase ofa strip with order.ratio[ ] equal to (0, 1, 2, 3,3,3,3).

In the preferred embodiment depicted in FIG. 15, after the loop hasiterated through all of the states (the state counter is incremented instep 1505) the loop terminates. After looping through all of the states,the limit order “price” of the “sale” must be revised so that it may beconverted into a complementary buy. This step is depicted in step 1506,where the revised limit order “price” for the complementary buy is equalto one minus the original limit order “price” for the “sell”. Afterfinishing the switching of each state in order.ratio[ ] and setting thelimit order “price” for each order, the loop which increments over theorders goes to the next order, as indicated in step 1507. The conversionof “sell” orders to buy orders terminates when all orders have beenprocessed as indicated in step 1508.

FIG. 16 depicts a preferred embodiment of a method for adjusting limitorders in the presence of transaction fees in a DBAR DOE of the presentinvention. The function which implements this embodiment isfeeAdjustOrders( ) and is invoked in the method for processing limit andmarket orders depicted and discussed with reference to FIG. 11. Limitorder are adjusted for transaction fees to reflect the preference thatorders (after all “sell” orders have been converted to buy orders)should only be executed when the trader specifies that he is willing topay the equilibrium “price,” inclusive of transaction fees. Theinclusion of fees in the “price” produces the “offer” price. Therefore,in a preferred embodiment, all or part of an order with a limit “price”which is greater than or equal to the “offer” price should be executedin the final equilibrium, and an order with a limit “price” lower thanthe “offer” price of the final equilibrium should not be executed atall. To ensure that this equilibrium condition obtains, in a preferredembodiment the limit order “prices” specified by the traders areadjusted for the transaction fee assessed for each order before they areprocessed by the equilibrium calculation, specifically the “add andprune” cycle discussed in Section 6 above and with reference to FIG. 17below, which involves the recomputation of equilibrium “prices.” Thus,the “add and prune” cycle is performed with the adjusted limit order“prices.”

Referring back to FIG. 16, which discloses the steps of the functionfeeAdjustOrders( ) step 1601 initiates a control loop for each order inthe contract (contract.numOrders). The next step 1602 queries whetherthe order being considered is a market order (order.marketLimit=1) or alimit order (order.marketLimit=0). A market order is unconditional andin a preferred embodiment need not be adjusted for the presence oftransaction fee, i.e., it is executed in full regardless of the “offer”side of the market. Thus, if the order is market order, its “limit”price or implied probability is set equal to one as shown in step 1604(order[j].limitPrice=1). If the order being processed in the controlloop of step 1601 is a limit order, then step 1603 revises the initiallimit order by setting the new limit order “price” equal to the initiallimit order “price” less the transaction fee (order.fee). Ina preferredembodiment, this function is called after all “sell” orders have beenconverted to buy orders, so that the adjustment for all orders mayinvolve only making the buy orders less likely to be executed byadjusting their respective limit “prices” down by the amount of the fee.After each adjustment is made, the loop over the orders is incremented,as shown in step 1605. After all of the orders have been processed, thefunction feeAdjustOrders( ) terminates as shown in step 1606.

FIG. 17 discloses a preferred embodiment of a method for filling oraddition and removal of lots in a DBAR DOE of the present invention. Thefunction fillRemoveLots( ) which is invoked in the central “add andprune” cycle of FIG. 11, is depicted in detail in FIG. 17. The functionfillRemoveLots( ) implements the method of binary search to determinethe appropriate number of lots to add (or “fill”) or remove, in thepreferred embodiment depicted in FIG. 17, lots are filled or added whenthe function is called with the first parameter equal to 1 and lots areremoved when the function is called with the first parameter equal tozero. The first step of function fillRemoveLots( ) is indicated in step1701. If lots are to be removed, then the method of binary search willtry to find the minimum number of lots to be removed such that the limit“price” of the order (order.limitPrice) is greater than or equal to therecalculated equilibrium “price” (order.price). Thus, if orders are tobe removed, step 1701 sets the maxPremium variable to the number of lotswhich are currently filled in the order, and sets the minPremiumvariable to zero. In other words, in preferred embodiments in a firstiteration the method of binary search will try to find a new number oflots somewhere on the interval between the currently filled number oflots and zero, so that the number of lots to be filled after the step iscompleted is the same or lower than the number of lots currently filled.If lots are to be filled or added, then the method of binary search setsthe maxPremium variable to the order amount (order.amount) since this isthe maximum amount that can be filled for any given order, and theminimum amount equal to the currently filled amount(minPremium=order.filled). That is, if lots are to be filled or added,the method of binary search will try to find the maximum number of lotsthat can be filled or added so that the new number of filled between thecurrent number of lots filled and the number of lots requested in theorder.

In the preferred embodiment depicted in FIG. 17, step 1702 bisects theintervals for binary search created in step 1701 by setting the variablemidPremium equal to the mid point of the interval created in step 1701.A calculation of equilibrium “prices” or implied probabilities for thegroup of DBAR contingent claims equilibrium calculation will then beattempted with the number of lots for the relevant orders reflected bythis midpoint, which will be greater than the current amount filled iflots are to be added and less than the current amount filled if lots areto be removed.

Step 1703 queries whether any change (to within a tolerance) in themid-point of the interval has occurred between the last and currentiteration of the process. If no change has occurred, a new order amountthat can be filled has been found and is revised in step 1708, and thefunction fillRemoveLots( ) terminates in step 1709. If the is differentfrom the midpoint of last iteration, then the new equilibrium iscalculated with the greater (in the case of addition) or lower (in thecase of removal) number of lots as specified in step 1702 of the binarysearch. In a preferred embodiment the equilibrium “prices” arecalculated with these new fill amounts by the multistate allocationfunction, compEq( ) which is described in detail with reference to FIG.13. After the invocation of the function compEq( ) each order will havea current equilibrium “price” as reflected in the data structure memberorder.price. The limit “price” of the order under consideration(order[j])) is then compared to the new equilibrium “price” of the orderunder consideration (order[j].price), as shown in step 1705. If thelimit “price” is worse, i.e., less than the new equilibrium or market“price,” then the binary search has attempted to add too many lots andtries again with fewer lots. The lesser number of lots with which toattempt the next iteration is obtained by setting the new top end of theinterval being bisected to the number of lots just attempted (whichturned out to be too large). This step is depicted in step 1706 of thepreferred embodiment of FIG. 17. With the interval thus redefined andshifted lower, a new midpoint is obtained in step 1702, and a newiteration is performed. If, in step 1705, the newly calculatedequilibrium “price” is less than or equal to the order's limit price,then the binary search will attempt to add or fill additional lots. Inthe preferred embodiment depicted in FIG. 17, the higher number of lotsto add is obtained in step 1707 by setting the lower end of the searchinterval equal to the number of lots for which an equilibriumcalculation was performed in the previous iteration. A new midpoint ofthe newly shifted higher interval is then obtained in step 1702, so thatthe another iteration of the search may be performed with a highernumber of lots. As previously indicated, once further iterations nolonger change the number of lots that are filled, as indicated in step1703, the number of lots of the current iteration is stored, asindicated in step 1708, and the function fillRemoveLots( ) terminates,as indicated in step 1709.

FIG. 18 depicts a preferred embodiment of a method of calculatingpayouts to traders in a DBAR DOE of the present invention, once therealized state corresponding to the event of economic significance orstate of a selected financial product is known. Step 1801 of FIG. 18shows that the predetermined termination criteria with respect to thesubmission of orders by traders have been fulfilled, for example, thetrading period has ended at a previous time (time=t) and the finalcontingent claim prices have been computed and finalized. Step 1802confirms that the event of economic significance or state of a financialproduct has occurred (at a later time=T, where T≧t) and that therealized state is determined to be equal to state k. Thus, according tostep 1802, state k is the realized state. In the preferred embodimentdepicted in FIG. 18, step 1803 initializes a control loop for each orderin the contract (contract.numOrders). For each order, the payout to thetrader is calculated. In preferred embodiments, the payout is a functionof the amount allocated to the realized state (order.invest[k]), theunit payout of the realized state(contract.totalInvested/contract.stateTotal[k]), and the transaction feeof the order as a percentage of the order price (order.fee/order.price).Other methods of allocating payouts net of transaction fees are possibleand would be apparent to one of ordinary skill in the art.

The foregoing detailed description of the figures, and the figuresthemselves, are designed to provide and explain specific illustrationsand examples of the embodiments of methods and systems of the presentinvention. The purpose is to facilitate increased understanding andappreciation of the present invention. The detailed description andfigures are not meant to limit either the scope of the invention, itsembodiments, or the ways in which it may be implemented or practiced.

In the embodiment described in Section 7, the DBAR DOE equilibrium iscomputed through a nonlinear optimization to determine the equilibriumexecuted amount for each order, x_(j), in terms of the notional payoutreceived should any state of the set of constituent states of a DBARdigital option occur (defined by B), such that limit orders can beaccepted and processed which are expressed in terms of each trader'sdesired payout (r_(j)). The descriptions of FIGS. 19 and 20 that followexplain this process in detail. Other aspects of this and otherembodiments of the present invention are depicted in FIGS. 21 to 25,referenced in Sections 3, 8 and 9 of this specification.

Generally speaking, in this embodiment, as described in Section 7, theDBAR DOE equilibrium executed amount for the orders is arrived at by:

-   -   (i) inputting into the system how many orders (n) and how many        states (m) are present in the contract;    -   (ii) for each order j, accepting specifications for order or        trade including: (1) if the order is a buy order or a “sell”        order; (2) requested notional payout size (r_(j)); (3) if the        order is market order or limit order; (4) limit order price        (w_(j)) (or if order is market order, then w_(i=1)); (5) the        payout profile or set of defined states for which desired        digital option is in-the money (row j in matrix B); and (6) the        transaction fee (f_(j)).    -   (iii) loading contract and order data structures;    -   (iv) placing opening orders (initial invested premium for each        state, k_(i));    -   (v) converting “sell” orders to complementary buy orders simply        by identifying the range of complementary states being “sold”        and, for each “sell” order j, adjusting the limit “price”        (w_(j)) to one minus the original limit “price” (1−w_(j));    -   (vi) adjusting the limit “price” to incorporate the transaction        fee to produce an adjusted limit price w_(j) ^(a) for each order        j;    -   (vii) grouping the limit orders by placing all of the limit        orders which span or comprise the same range of defined states        into the same group;    -   (viii) sorting the orders upon the basis of the limit order        “prices” from the best (highest “price” buy) to the worst        (lowest “price” buy);    -   (ix) establishing an initial iteration step size, α_(i)(1), the        current step size, α_(j)(κ), will equal the initial iteration        step size, α_(j)(1), until and unless adjusted in step (xii);    -   (x) calculating the equilibrium to obtain the total investment        amount T and the state probabilities, p's, using Newton-Raphson        solution of Equation 7.4.1(b);    -   (xi) computing equilibrium order prices (π_(j)'s) using the p's        obtained in step (viii);    -   (xii) incrementing the orders (x_(j)) which have adjusted limit        prices (w_(j) ^(a)) greater than or equal to the current        equilibrium price for that order (π_(j)) from step (ix) by the        current step size α_(j)(κ);    -   (xiii) decrementing the orders (x_(j)) which have limit prices        (w_(j)) less than the current equilibrium price for that order        (π_(j)) from step (ix) by the current step size α_(j)(κ);    -   (xiv) repeating steps (ix) to (xii) in subsequent iterations        until the values obtained for the executed order notional        payouts achieve a desired convergence, adjusting the current        step size α_(j)(κ) and/or the iteration process after the        initial iteration to further progress towards the desired        convergence;    -   (xv) achieving a desired convergence (along with a final        equilibrium of the prices p's and the total premium invested T)        of the maximum executed notional payout orders x_(j) when        predetermined convergence criteria are met;    -   (xvi) taking the “prices” resulting from the solution final        equilibrium resulting from step (xiii) and adding any applicable        transaction fee to obtain the offer “price” for each respective        contingent claim ordered and subtracting any applicable        transaction fee to obtain the bid “price” for each respective        contingent claim ordered; and    -   (xvii) upon fulfillment of all of the termination criteria        related to the event of economic significance or state of a        selected financial product, allocating payouts to those orders        which have investments on the realized state, where such payouts        are responsive to the final equilibrium “prices” of the orders'        contingent claims and the transaction fees for such orders.

Referring to FIG. 19, illustrative data structures are depicted whichmay be used to store and manipulate the data relevant to the DBAR DOEembodiment described in Section 7 (as well as other embodiments of thepresent invention): data structures for a “contract” (1901), for a“state” (1902) and for an “order” (1903). Each data structure isdescribed below, however it is understood that depending on the actualimplementation of the stepping iterative algorithm, different datamembers or additional data members may be used to solve the optimizationproblem in 7.7.1.

The data structure for a “contract” or group of DBAR contingent claims,shown in 1901, includes data members which store data which are relevantto the construction of the DBAR DOE contract or group of claims underthe embodiment described in Section 7 (as well under other embodimentsof the present invention). Specifically, the contract data structureincludes the following members (also listing the variables denoted bysuch members as described above, if any, and proposed member names forlater programming the stepping iterative algorithm):

-   -   (i) the number of defined states i (m, contract.numStates);    -   (ii) the total premium invested in the contract (T,        contract.totalInvested);    -   (iii) the number of orders j (n, contract.numOrders);    -   (iv) a list of the orders and each order's data (contract.orders        [ ]); and    -   (v) a list of the states and each state's data (contract.states        [ ]).

The data structure for a “state”, shown in 1902, includes data memberswhich store data which are relevant to the construction of each DBAR DOEstate (or spread or strip) under the embodiment described in Section 7,as well as under other embodiments of the present invention.Specifically, each state data structure includes the following members(also listing the variables denoted by such members as described above,if any, and proposed member names for later programming the steppingiterative algorithm):

-   -   (i) the total premium invested in each state i (T_(i),        state.stateTotal);    -   (ii) the executed notional payout per defined state i (y_(i),        state.poReturn[ ]);    -   (iii) the price/probability for each state i (p_(i),        state.statePrice); and    -   (iv) the initial invested premium for each state i to initialize        the contract (k_(i), state.initialState).

The data structure for an “order”, shown in 1903, includes data memberswhich store data which are relevant to the construction of each DBAR DOEorder under the embodiment described in Section 7, as well as underother embodiments of the present invention. Specifically, each orderdata structure includes the following members (also listing thevariables denoted by such members as described above, if any, andproposed member names for later programming the stepping iterativealgorithm):

-   -   (i) the limit price for each order j (w_(j), order.limitPrice);    -   (ii) the executed notional payout per order j, net of fees,        after all predetermined termination criteria have been met        (x_(j), order.executedPayout);    -   (iii) the equilibrium price/probability for each order j (π₁,        order.orderPrice);    -   (iv) the payout profile for each order j (row j of B,        order.ratio[ ]), specifically it is a vector which specifies the        type of contingent claim to be traded (order.ratio[ ]). For        example, in an embodiment involving a contract with seven        defined states, an order for a digital call option which would        expire in the money should any of the last four states occur        would be rendered in the data member order.ratio[ ] as        order.ratio[0,0,0,1,1,1,1] where the 1's indicate that the same        payout should be generated by the multistate allocation process        when the digital option is in the money, and the 0's indicate        that the option is out of the money, or expires on one of the        respective out of the money states. As another example, a spread        which is in the money should states either states 1,2, 6, or 7        occur would be rendered as order.ratio[1,1,0,0,0,1,1]. As        another example, a digital option strip, which allows a trader        to specify the relative ratios of the final payouts owing to an        investment in such a contingent claim would be rendered using        the ratios over which the strip is in the money. For example, if        a trader desires a strip which pays out three times much as        state 3 should state 1 occur, and twice as much as state 3 if        state 2 occurs, the strip would be rendered as        order.ratio[3,2,1,0,0,0,0]. In other words, the vector stores        integers which indicate the range of states in which an        investment is to be made in order to generate the payout profile        of the contingent claim desired by the trader placing the order.    -   (v) the transaction fee for each order j (f_(j), order.fee);    -   (vi) the requested notional payout per order j (r_(j),        order.requestedPayout);    -   (vii) whether order j is a limit order whose viability for        execution is conditional upon the final equilibrium “price”        being below the limit price after all predetermined termination        criteria have been met, or whether order j is a market order,        which is unconditional (order.marketLimit=0 for a limit order,        =1 for a market order);    -   (viii) whether order j is a buy order or a “sell” order        (order.buySell=1 for a buy, and =−1 for a “sell”); and    -   (ix) the difference between market price and limit price per        order j order.priceGap).

FIG. 20 depicts a logical diagram of the basic steps for limit andmarket order processing in the embodiment of a DBAR DOE described inSection 7, which can be applied to other embodiments of the presentinvention. Step 2001 of FIG. 20 inputs into the system how many orders(contract.numOrders) and how many states (contract.numStates) arepresent in the contract. Then, in step 2002, the computer system acceptsspecifications from the trader or user for each order, including: (1) iforder is a buy order or a “sell” order (order.buySell); (2) requestednotional payout size (order.requestedPayout); (3) if order is marketorder or limit order (order.marketLimit); (4) limit order price(order.limitPrice) (or if order is market order, thenorder.limitPrice=1); (5) the payout profile or set of defined states forwhich desired digital option is in-the money (order.ratio[ ]); and (6)transaction fee (order.fee).

Step 2003 of FIG. 20 loads the relevant data into the contract, stateand order data structures of FIG. 19. The initial value oforder.executedPayout and state.poReturn are set at zero.

Step 2004 initializes the set of DBAR contingent claims, or the“contract,” by placing initial amounts of value units (i.e., initialliquidity) in each state of the set of defined states. The placement ofinitial liquidity avoids a singularity in any of the defined states(e.g.; an invested amount in a given defined state equal to zero) whichmay tend to impede multistate allocation calculations. Theinitialization of step 2004 may be done in a variety of different ways.In this embodiment, a small quantity of value units is placed in each ofthe defined states. For example, a single value unit (“lot”) may beplaced in each defined state where the single value unit is expected tobe small in relation to the total amount of volume to be transacted. Instep 2004 of FIG. 20, the initial value units are represented in thevector init[contract.numStates].

In this embodiment, step 2005 of FIG. 20 invokes the functionadjustLimitPrice( ) which converts the limit order price of the “sell”orders to the limit order price of complementary buy orders, and adjuststhe limit order prices to account for the transaction fee charged forthe order (subtracting the fee from the limit order price for a buyorder and subtracting the fee from the converted limit order price for a“sell” order). After the completion of step 2005, all of the limit orderprices for contingent claims—whether buy or “sell” orders, can beprocessed as buy orders together, and the limit order prices areadjusted with fees for the purpose of the ensuing equilibriumcalculations.

In this embodiment, step 2006 groups these buy orders based upon thedistinct ranges of states spanned by the contingent claims specified inthe orders. The range of states comprising the order are contained inthe data member order.ratio[ ] of the order data structure 1903 depictedin FIG. 19. As with the DBAR DOE embodiment discussed in section 6 andFIG. 12 above and other embodiments of the present invention, eachdistinct order[j].ration[ ] vector in step 2006 in FIG. 20 is groupedseparately from the others in step 2006. Two order[j].ratio[ ] vectorsare distinct for different orders when their difference yields a vectorthat does not contain zero in every element. For example, for a contractwhich contains seven defined states, a digital put option which spansthat first three states has an order[1].ratio[ ] vector equal to(1,1,1,0,0,0,0). A digital call option which spans the last five stateshas an order[2].ratio[ ] vector equal to (0,0,1,1,1,1,1). Because thedifference of these two vectors is equal to (1,1,0,−1,−1,−1,−1), thesetwo orders should be placed into distinct groups, as indicated in step2006.

In this embodiment, step 2006 aggregates orders into relevant groups forprocessing. For the purposes of processing limit orders: (i) all ordersmay be treated as limit orders since orders without limit “price”conditions, e.g., “market orders,” can be rendered as limit buy orders(including “sale” orders converted to buy orders in step 2005) withlimit “prices” of 1, and (ii) all order sizes are processed by treatingthem as multiple orders of the smallest value unit or “lot.”

The relevant groups of step 2006 of FIG. 20 are termed “composite” sincethey may span, or comprise, more than one of the defined states. Forexample, the MSFT Digital Option contract depicted above in Table 6.2.1has defined states (0,30], (30,40], (40,50], (50,60], (60, 70], (70,80], and (80,00]. The 40 strike call options therefore span the fivestates (40,50], (50,60], (60, 70], (70, 80], and (80,00]. A “sale” of a40 strike put, for example, would be aggregated into the same group forthe purposes of step 2004 of FIG. 20, because the “sell” limit order ofa 40 strike put has been converted at step 2005 into a complementary buyorder of a 40 strike call simply by converting the limit order price forthe put order into the complementary limit order price of the callorder.

Similar to step 1206 of DBAR DOE embodiment described with reference toFIG. 12, at the point of step 2007 of this embodiment shown in FIG. 20,all of the orders may be processed as buy orders (because any “sell”orders have been converted to buy orders in step 2005 of FIG. 20) andall limit “prices” have been adjusted (with the exception of marketorders which, in an embodiment of the DBAR DOE or other embodiments ofthe present invention, have a limit “price” equal to one) to reflecttransaction costs equal to the fee specified for the order's contingentclaim (as contained in the data member order[j].fee).

In this embodiment, step 2007 sorts each group's orders based upon theirfee-adjusted limit “prices,” from best (highest “prices”) to worst(lowest “prices”). The grouped orders follow the same aggregation asillustrated in Diagram 1 above, and in Section 6. Step 2008 establishesan initial iteration step size, init[order.stepSize], the current stepsize, order.stepSize, will equal the initial iteration step size untiland unless adjusted in step 2018.

Initially as part of a first iteration (numIteration=1) (2009 a), andlater as part of subsequent iterations, step 2009 invokes the functionfindTotal( ) which calculates the equilibrium of Equation 7.4.7 toobtain the total investment amount (contract.totalInvested) and thestate probabilities (state.statePrice). Step 2010 invokes the functionfindOrderPrices( ) which computes the equilibrium order prices(order.orderPrice) using the state probabilities (state.statePrice)obtained in step 2009. The equilibrium order price for each order(order.orderPrice) is equal to the payout profile for the order(order.ratio[ ]) multiplied with a vector made up of the probabilitiesfor all states i (state.statePrice[contract.numStates]).

Proceeding with the next step of this embodiment depicted in FIG. 20,step 2011 queries whether there is at least a single order which has alimit “price” which is “better” than the current equilibrium “price” forthe ordered option. In this embodiment, for the first iteration of step2011 for a trading period for a group of DBAR contingent claims, thecurrent equilibrium “prices” reflect the placement of the initialliquidity from step 2004. Step 2012 invokes the incrementing( )function, which increments the executed notional payout(order.executedPayout) with the current step size (order.stepSize) foreach order which has a limit price (order.limitPrice) greater than orequal to the current equilibrium price for that order (order.orderPrice)obtained from step 2010 (however, in this embodiment, such incrementingshould not exceed the order's requested payout r_(j)).

Similarly, step 2013 queries whether there is at least a single orderwhich has a limit “price” which is “worse” than the current equilibrium“price” for the ordered option. Step 2014 invokes the decrementing( )function, which decrements the executed notional payout(order.executedPayout) with the current step size (order.stepSize) foreach order which has a limit price (order.limitPrice) less than thecurrent equilibrium price for that order (order.orderPrice) obtainedfrom step 2010 (but, in this embodiment, such decrementing should notproduce an executed order payout below zero).

This embodiment of the DBAR DOE (described in Section 7) simplifies thecomplex comparison and removes the necessity of the “add” and “prune”method for buy and “sell” orders in the DBAR DOE embodiment described inSection 6. In this embodiment (depicted in FIG. 20), once the limitorder price for “sell” orders has been converted to a complementarylimit order price for a buy order, with both types of orders alreadybeing expressed in terms of payout, the notional payout executed foreither a buy or a “sell” order (order.executedPayout) is simplyincremented by the current step size (order.stepSize) if the limit orderprice (order.limitPrice) is greater than or equal to the currentequilibrium price (order.orderPrice), and decremented by the currentstep size (order.stepSize) if the limit order price (order.limitPrice)is less than the current equilibrium price (order.orderPrice).

In step 2015, the counter for the iteration (numIteration) isincremented by 1. Repeat steps 2009 to 2014 for a second iteration(until numIteration=3). Step 2016 queries whether the quantitiescalculated for the executed notional payouts for the orders(order.executedPayout) are converging, and whether the convergence needsto be accelerated. If the executed notional payouts calculated in 2014are not converging or the convergence needs to be accelerated, step 2017queries if the step size (order.stepSize) needs to be adjusted. If thestep size needs to be adjusted, step 2018 adjusts the step size(order.stepSize). Step 2019 queries if the iteration process needs toaccelerated. Step 2020 initiates a linear program if the iterationprocess needs to be accelerated. Then, the iteration process (steps 2009to 2014) is repeated, again. However, if after step 2016, the quantitiescalculated for the executed notional payouts for the orders(order.executedPayout) have converged (according to some possiblypredetermined or dynamically determined convergence criteria), then theiteration process is complete, and the desired convergence has beenachieved in step 2021, along with a final equilibrium of the orderprices (order.orderPrice) and total premium invested in the contract(contract.totalInvested), and determination of the maximum executednotional payouts for the orders (order.executedPayout).

In step 2022, the order price, not including transaction fees, iscalculated by adding any applicable transaction fee (order.fee) to theequilibrium order price (order.orderPrice) to produce the equilibriumoffer price, and subtracting any applicable transaction fee (order.fee)to the equilibrium order price (order.orderPrice) to produce theequilibrium bid price.

In step 2023, upon fulfillment of all of the termination criteriarelated to the event of economic significance or state of a selectedfinancial product, allocating payouts to those orders which haveinvestments on the realized state, where such payouts are responsive tothe final equilibrium “prices” of the orders' contingent claims(order.orderPrice) and the transaction fees for such orders (order.fee).

The steps and data structures described above and shown in FIGS. 11 to25 for embodiments of DBAR digital options (discussed, for example, inSections 6 and 7 herein) and an embodiment of a demand-based market orauction for structured financial products (discussed, for example, inSection 9 herein), can be implemented within the computer systemdescribed above in reference to FIGS. 1 to 10, as well as in otherembodiments of the present invention. The computer system can includeone or more parallel processors to run, for example, the linear programfor the optimization solution (Section 7), and/or to run one or morefunctions in the DRF or OPF in parallel with a main processor in theacceptance and processing of any DBAR contingent claims, includingdigital options. For DBAR digital options, in addition to determiningand allocating a payout at the end of the trading period, the trader oruser or investor specifies and inputs a desired payout, a selectedoutcome and a limit order price (if any) into the system during thetrading period, and the system determines the investment amount for theorder at the end of the trading period along with an allocation ofpayouts. In other words, the processor and other components (includingcomputer usable medium having computer readable program code, andcomputer usable information storage medium encoded with acomputer-readable data structure) causes the computer system to acceptinputs of information related to a DBAR digital option or to other DBARcontingent claims, perhaps by way of a propagated signal or from aremote terminal by way of the Internet or a private network withdedicated circuits, including each trader's identity, and the desiredpayout, payout profile, and limit price for each order, then throughoutthe trading period the computer system updates the allocation of payoutsper order and the investment amounts per order, and communicates theseupdated amounts to the trader (and, in the case of other DBAR contingentclaims, inputted information may include the investment amount so thatthe computer system can allocate payouts per defined state). At the endof the trading period, the computer system determines a finalizedinvestment amount per order (for DBAR digital options) and allocation ofpayouts per order if the states selected in the order become the statescorresponding to the observed outcome of the event of economicsignificance. In the above DBAR digital option embodiments, the ordersare executed after the end of the trading period at these finalizedamounts. The determination of the investment amount and payoutallocation can be accomplished using any of the embodiments disclosedherein, alone or in combination with each other.

Additionally, the implementations in a computer system (or with anetwork implementation) of the methods described herein to determine theinvestment amount and payout allocation as a function of the desiredpayout, selected outcomes, and limit order prices for each order placedin a DBAR digital options market or auction (or to determine the payoutas a function of the selected outcomes, and investment amounts for eachorder in other embodiments of DBAR contingent claims), can be used by abroker to provide financial advice to his/her customers by helping themdetermine when they should invest in a DBAR digital options market orauction based on the type of return they would like to receive, theoutcomes they would like to select, and the limit order price (if any)that they would like to pay or if they should invest in another DBARcontingent claim market or auction based on the amount they would liketo invest, their selected outcomes and other information as describedherein.

Similarly, the implementations and methods described herein can be usedby an investor as a method of hedging for any of the types of economicevents (including any underlying economic events or measured parametersof an underlying economic event as discussed above, including Section3). Hedging involves determining an investment risk in an existingportfolio (even if it includes only one investment) or determining arisk in an asset portfolio (a risk in a lower farm output due to badweather, for example), and offsetting that risk by taking a position ina DBAR digital option or other DBAR contingent claim that has anopposing risk. On the flip side, if a trader is interested in increasingthe risk in an existing portfolio of investments or assets, the DBARdigital option or other DBAR contingent claim is a good tool forspeculation. Again, the trader determines the investment risk in theirasset or investment portfolio, but then takes a position in DBAR digitaloption or other DBAR contingent claim with a similar risk.

The DBAR digital option described above is one type of instrument fortrading in a demand-based market or auction. The digital option setsforth designations of information which are the parameters of the option(like a coupon rate for a Treasury bill), such as the payout profile(corresponding to the selected outcomes for the option to bein-the-money), the desired payout of the option, and the limit orderprice of the option (if any). Other DBAR contingent claims describedabove are other types of instruments for trading in a demand-basedmarket or auction. They set forth parameters including the investmentamount and the payout profile. All instruments are investment vehiclesproviding investment capital into a demand-based market or auction inthe manner described herein.

The replication of derivatives strategies and financial products, andthe enablement of trading derivatives strategies and financial productsin demand-based markets or auctions shown in FIGS. 26, 27A-27C and28A-28C (discussed, for example, in Section 10 herein), can also beimplemented within the computer system described above in reference toFIGS. 1 to 25, as well as in other embodiments of the present invention.The computer system can include one or more parallel processors to run,for example, a replication solution for derivatives strategies orfinancial products, and/or to run one or more functions in the DRF orOPF in parallel with a main processor in the acceptance and processingof any replicated derivatives strategies, financial products and DBARcontingent claims, including digital options. The processor and othercomponents (including computer usable medium having computer readableprogram code, and computer usable information storage medium encodedwith a computer-readable data structure) causes the computer system toaccept inputs of information related to a replicated derivativesstrategy and/or to DBAR contingent claims, perhaps by way of apropagated signal or from a remote terminal by way of the Internet or aprivate network with dedicated circuits, including each trader'sidentity, one or more parameters of a derivatives strategy and/orfinancial product in each order, then throughout the trading period thecomputer system updates the allocation of payouts and prices orinvestment amounts per order, and communicates these updated amounts tothe trader. At the end of the trading period, the computer systemdetermines a finalized investment amount per order (for replicatedderivatives strategies and/or financial products) and allocation ofpayouts per order if the states selected in the order become the statescorresponding to the observed outcome of the event of economicsignificance. In the above replicated derivatives strategy and/orfinancial product embodiments, the orders are executed after the end ofthe trading period at these finalized amounts. The determination of theinvestment amount and payout allocation for each contingent claim in thereplication set for the derivatives strategy and/or financial productcan be accomplished using any of the embodiments disclosed herein, aloneor in combination with each other.

The implementations in a computer system (or with a networkimplementation) of the methods described herein to determine theinvestment amounts and payout allocation for replicated derivativesstrategies and/or financial products, can also be used by a broker toprovide financial advice to his/her customers by helping them determinewhen they should invest in a derivatives strategy and/or financialproduct in a demand-based market or other type of market based on thetype of return they would like to receive, the outcomes they would liketo select, and the limit order price (if any) that they would like topay or the amount they would like to invest for the derivatives strategyand/or financial product, and other information as described herein.

The implementations and methods described herein can also be used by aninvestor as a method of hedging for any of event (including anyunderlying event or measured parameters of an underlying event asdiscussed above, including Section 10). Hedging involves determining aninvestment risk in an existing portfolio (even if it includes only oneinvestment) or determining a risk in an asset portfolio, and offsettingthat risk by taking a position in a replicated derivatives strategyand/or financial product that has an opposing risk. On the flip side, ifa trader is interested in increasing the risk in an existing portfolioof investments or assets, the replicated derivatives strategy and/orfinancial product is a good tool for speculation. Again, the traderdetermines the investment risk in their asset or investment portfolio,but then takes a position in a replicated derivatives strategy and/orfinancial product with a similar risk.

13. DBAR SYSTEM ARCHITECTURE (AND THE DETAILED DESCRIPTION OF THEDRAWINGS IN FIGS. 32 TO 68) 13.1 Terminology and Notation

The following terms shall have the meanings set forth below:

-   -   Auction—DBAR auction.    -   Event—Underlying event for a DBAR auction.    -   User—Someone who accesses the system using a web browser.    -   Group—All customer and administrator users must belong to a        group; members of a group are allowed to view and modify each        other's orders.    -   Desk—the system configuration.    -   State—the term “state” as used in this section means the        condition of being or phase for a given auction and is different        from the meaning of “state” in previous sections.    -   Transaction—there can be four types of transactions in the        system: auction, event, user and group. Generally when a        transaction is not qualified with one of these, it is assumed to        be an auction transaction. Each are defined in more detail        below:        -   Auction transaction—these are of the type: auction            configuration (add, replace), order (add/modify/cancel),            state change (open, close, cancel, finalize) or final            report. All events for an auction are kept together in a            directory and are numbered sequentially.        -   Event transaction—create event. They are deleted very            carefully when the system is offline to prevent referential            integrity issues.        -   User transactions—create/modify user.        -   Group transactions—create/modify group.            The notation used for flowcharts and pseudo-code is defined            in the legend in section 13.12.1

13.2 Overview

This document describes an example embodiment of an electronic DBAR (ordemand-based) trading system used to provide a system implementation ofthe embodiment described in Section 11. This example embodiment of anelectronic DBAR trading system can also be used with respect to theother embodiments of a DBAR auction described in this specification. Inthis example embodiment, users access the system over the internethrough the https protocol using commonly available web browsers. Thesystem provides for three types of users:

-   -   Public—can view auction information, prices and distributions        without logging in.    -   Customers—can login and view auction information, prices,        distribution, and order summary, and can place orders.    -   Administrators—can create users, events, groups and DBAR        auctions, control auctions, and request auction reports to be        used for order execution at the end of an auction.

The system provides users with real-time pricing and order fill updatesas new orders are received. The system is designed to minimize the timebetween when a new order is accepted and when the pricing and orderfills that reflect the effect of that order are available for display toall users. This time is typically less than five seconds. This isachieved through the use of a fast in-memory database and a highlyoptimized implementation of the core algorithm. A key element inproviding this rapid level of response is that the core algorithm mustrun on the fastest available microprocessor, in particular one that hasexcellent floating point performance such as the Intel Pentium 4. Sincethe algorithm is compute-bound, typically an entire processor isdedicated for equilibrium calculations.

The system also guarantees that the prices and order fills do notviolate the parimutuel equilibrium by more than a system specified(economically insignificant) amount. This is done by taking the resultsof an equilibrium calculation and checking it using ‘run-timeconstraints’, confirming that the mathematical requirements of thesystem are met, as summarized in section 11.4. These constraints checkthat:

-   -   the prices of the replication claims are positive (equation        11.4.3A) and sum to one (equation 11.4.3B);    -   option prices are weighted sums of the replication prices        (equation 11.4.3C);    -   the limit price logic is met for buys and sells (equation        11.4.4C); and    -   the self-hedging condition of equation 11.4.5E is met for all        outcomes of the underlying U to within either a 1,000 currency        units or 1 basis point of replicated premium M (defined in        equation 11.4.5A).

The system will not publish prices and fills that do not meet theseconstraints, thus assuring that the user will not see prices which areinvalid due to a calculation error.

The system also provides a “limit order book” for each option in anauction. This feature is unique to this type of marketplace andimplementation in that it provides the user with real-time informationon the fill volume that the system would provide in buying or selling anoption at a price above or below the current market price. While thisseems much like the traditional limit order book in a continuouslytraded market, it is different in that the system does not requireproposed sells to match with existing buys or proposed buys to bematched with existing sells of the same option.

The system runs using a highly redundant network of servers whose accessto the interne is controlled by firewalls. Specifically, the system isredundant in the following ways:

-   -   Geographically diverse and redundant data centers;    -   Multiple connections to the internet using multiple carriers;    -   All network devices are redundant eliminating all single points        of failure with automatic failover; and    -   All servers are redundant eliminating all single points of        failure with a combination of automatic and manual failover.

13.3 Application Architecture

The application is implemented as a collection of processes which areshown in FIG. 32. This figure also details their locations, and themessage flow between processes.

The processes communicate via a messaging middleware product such as PVM(open source—http://www.csm.ornl.gov/pvm/pvm_home.html) or TibcoRendezvous(http://www.tibco.com/solutions/products/active_enterprise/default.jsp).

13.3.1 uip 3202 (User Interface Processor)

The process uip 3202 is responsible for handling all web (https)requests from users' browsers to the system. These processes are spawnedby fastcgi running under apache web server (http://www.apache.org/).There are multiple processes per server and multiple servers per system.

13.3.2 ap 3206 (Auction Processor)

The process ap 3206 is responsible for:

-   -   Processing or delegating all event configuration, auction        configuration, state changes and orders from uip's 3202;    -   Handling state changes and event configuration itself (see more        details below), orders and auction configuration are passed to        the appropriate ce 3216;    -   Writing all valid requests to disk, then putting them in db        3208;    -   Notifying ce 3216 when auction transactions (orders or        configuration changes) have been put into db 3208, by sending        the last sequence number down;    -   On startup, restoring events, loading auction transaction to db        3208; and    -   Starting ce 3216, if not started for an auction

In order to achieve sufficient throughput to handle many (50-100)simultaneous auctions, ap 3206 is written to perform a minimal amount ofwork and to not wait on other processes. The system can be scaled byadding multiple ap 3206 processes if it is necessary to run moreauctions than a single ap 3206 can handle.

13.3.3 db 3208 (Database)

The process db 3208 is a fast in-memory object database used to storeand access all information used by the system.

Specifically it holds the following information:

-   -   auctions    -   events    -   users    -   user groups    -   desk    -   orders reports for auctions    -   prices/fills reports for auctions    -   internal reports used by the le 3218 for speeding up the        calculation of limit order book points    -   auction transactions (orders, configuration changes and state        changes)

13.3.4 dp 3210 (Desk Processor)

The process dp 3210 is responsible for restoring and configuring users,user groups, and the desk.

13.33 ce 3216 (Calculation Engine)

The process ce 3216 is responsible for performing the fundamentalequilibrium calculations that result in the prices and order fills foran auction at any given point in time.

It does not do any disk I/O and is stateless. If restarted, it gets allthe information it needs from db 3208 and recalculates a newequilibrium. This is to minimize the impact if the server it is runningon fails or if the calculation itself fails and must be restarted on adifferent server.

It also checks the semantics on all orders and rejects any that fail.This allows the auction administrator to change the strikes at any timewith the caveat that they may be causing orders to be rejected.

13.3.6 lp 3212 (Limit Order Book [LOB] Processor)

The process lp 3212 is responsible for accepting limit order bookrequests from the uip 3202 and then assigning the request to the nextavailable le 3218 for processing. If no le 3218 is available the requestis queued and then processed as soon as an le 3218 becomes available.

This process also watches to make sure that an le 3218 which has beenassigned a request always returns, and if it times out, then the requestis assigned to a new le 3218.

13.3.7 le 3218 (Limit Order Book [LOB] Engine)

The process le 3218 is responsible for calculating a set of limit orderbook prices requested by the uip 3202 for a specific option during anauction.

This calculation is similar to what the ce 3216 does, but it is donerepetitively at different limit prices for a set of hypothetical “whatif” orders. The purpose is to generate a set of fills that could berealized if a user were to buy options above the current market price orsell options below the current market price.

This calculation is sped up significantly by using the results of thelast equilibrium calculation (from the ce 3216) as a starting pointwhich is retrieved from the db 3208 at the start of each calculation.

13.3.8 resd 3204 (Resource Daemon)

The process resd 3204 is responsible for starting all other processesand monitoring their behavior and existence.

If resd 3204 determines that a process it is watching has exited orfailed, it will restart the process and then notify all other processesthat interact with the restarted process of the new process. This allowsthe system to continue operation with minimal user impact when a processfails to respond or exits abnormally.

13.3.9 logd 3214 (Logging Daemon)

The process logd 3214 is responsible for writing all log messages thatit receives from all other processes in the system. It serves as acentral point of logging and allows efficient monitoring of applicationstates and errors over a network connection.

13.4 Data

The system acts on a set of fundamental data types (or objects) definedbelow. Each of these is entered through the user interface with theexception of the desk, which is configured prior to starting the system.

Descriptions of the individual elements are contained in Appendix 13A.

13.4.1 Desk

The desk elements are as follows:

-   -   revision    -   revisionDate    -   revisionBy    -   desk    -   sponsor    -   sponsorld    -   limitOffsets

13.4.2 Users

Users are created or modified by administrative users. Each userrepresents a unique entity that can login to the system through theweb-based user interface to view auction information and place orders.

The user elements are as follows:

-   -   revision    -   revisionDate    -   revisionBy    -   userId    -   is Deleted    -   pswChangedDate    -   lastName    -   firstName    -   phone    -   email    -   location    -   description    -   groupId    -   canChangePsw    -   mustChangePsw    -   pswChangeInterval    -   accessPrivileges    -   loginId    -   password    -   failedLogins

13.4.3 Groups

Groups are created or modified by administrative users. Groups provide amechanism for administrators to group users who can view each other'sorders.

The group elements are as follows:

-   -   revision    -   revisionDate    -   revisionBy    -   groupName    -   groupId    -   is Deleted

13.4.4 Events

Events are created by administrative users. They can only becreated—never modified—to maintain referential integrity.

The event elements are as follows:

-   -   eventId    -   eventSymbol    -   eventDescription    -   currency    -   strikeUnits    -   expiration    -   tickSize    -   tickValue    -   floor    -   cap    -   payoutSettlementDate

13.4.5 Auctions

Auctions are created or modified by administrative users. Auctions havefour states, which are described below in the section “Auction State.”

The auction elements are as follows:

-   -   revision    -   revisionDate    -   revisionBy    -   auctionId    -   eventId    -   auctionSymbol    -   title    -   abstract    -   start    -   end    -   state    -   premiumSettlement    -   digitalFee    -   vanillaFee    -   digitalComboFee    -   vanillaComboFee    -   forwardFee    -   marketMakingCapital    -   strikes    -   openingPrices    -   vanillaPricePrecision

13.4.6 Orders

The order elements are as follows:

-   -   revision    -   revisionDate    -   revisionBy    -   orderId    -   groupId    -   optionType    -   lowerStrike    -   upperStrike    -   revision    -   revisionDate    -   revisionBy    -   is Canceled    -   side    -   limitPrice    -   amount    -   fill    -   mktPrice    -   userId    -   premiumCustomerReceives    -   premiumCustomerPays

13.5 Auction and Event Configuration 13.5.1 Auction Configuration

Auctions can be created or modified. When an auction is modified anyelement may be changed except the system assigned auctionId. If anauction element is changed (such as the removal of a strike) whichinvalidates orders in the system, then those orders will be marked witha status of rejected, which effectively means that the system hascanceled the order.

13.5.2 Event Configuration

Events can only be created—they cannot be modified or deleted using theuser interface. This is to maintain referential integrity. Events thatare no longer needed will be deleted using an offline utility when ithas been determined that they have no further references. If an event isincorrect, the administrator must create a new one and modify anyauctions to point at it.

13.6 Order Processing 13.6.1 Order Processing (Orders Placed byCustomers)

An order is received by the uip 3202 and subjected to a semantic andsyntactic check. Assuming it passes, it is then sent via pvm to the ap3206. It should return quickly with an indication of whether there wasan error. Generally the only error seen here would be if the auctionwere closed while the order was in flight.

ap 3206 first checks to see if the auction is open. If not, the order isnot accepted (error back to the uip 3202). Note that orders fromadministrators have a different type that allows the ap 3206 to easilydetermine if it can accept them without looking inside them.

The order is written to disk and to db 3208. ap 3206 sends a message toce notifying it of new orders. This message includes the sequence numberof the order (same one used in writing it to disk).

When the ce receives the notification of a new order, it requests thelatest orders from db 3208. Note that this request must at least provideorders up to and including the sequence number of the notify and mayalso include later orders.

ce 3216 checks all new orders for errors, and any that have errors aremarked as order status rejected. It is generally very unlikely that anyerrors would be found here but if they were, it would probably be due toone of two problems: (1) the system has some sort of bug that allowedthe error to get through the uip 3202, but failed here, or the systemmangled the order in transit, or (2) (more likely) the auctionadministrator changed the auction configuration, thereby eliminating thestrike on which this order was placed.

All orders are now used to produce a new orders report that is stored indb 3208. At this point the new order(s) is available for uip's 3202 andwill be visible in the user interface as a pending order.

A new equilibrium is calculated and checked with the constraints. If theconstraints fail, then system error messages are logged and operatorsmust correct the problem.

The new prices/fill report and ce 3216 report are stored in db 3208. Atthis point the order has been included in the current pricing and willbe marked as active in the user interface.

ce 3216 will reply to the “new transaction” message from the ap 3206with the last sequence number that was included in this equilibrium,letting it know that the transactions were successfully processed.

13.6.2 Order Processing (Orders Placed by Administrators)

This is identical to customers except that orders are never refused atthe ap 3206 since they can place orders while the auction is both openand closed.

13.6.3 Order Processing at Restore or Restart of a ce

A ce 3216 does not know the difference between the first time it isstarted and when it is restarted due to some sort of error. A ce 3216also does not know the state of the auction.

When a ce 3216 starts there are two possibilities—either there arecurrent valid reports for the auction or they do not exist yet. In anyevent, the ce 3216 does not delete any reports for an auction but ratherwill always overwrite them with newly calculated reports.

The ce 3216 looks for the latest auction configuration transaction anduses it to write (overwrite) the auction to db 3208.

The ce 3216 reads all order transactions and uses them to calculate anequilibrium, and then writes (overwrites) all reports into db 3208.

At this point the auction is fully restored and started in a consistentmanner.

13.7 Auction State

All auctions in the system have an attribute “state” which determines:

-   -   whether orders are accepted, and from whom;    -   if LOBs are displayed; and    -   if the auction has been completed.

Specifically there are four auction states, defined as follows:

-   -   Open 3304—accept orders from customers and administrators.    -   Closed 3302—allow only orders from administrators.    -   Canceled 3308—do not allow any orders, do not display        information on the auction (except that it has been canceled).    -   Finalized 3306—the auction is complete; do not allow any orders,        present the administrators with the ability to download a final        report that contains the final pricing and orders, fills, and        disposition of all orders.

Auctions begin in the closed 3302 state and follow a specific transitionsequence as shown in FIG. 33.

13.7.1 State Changes

Only administrators can request state changes to the auction.

ap 3206 writes state change requests to disk as an auction transactionand then to db 3208. Based on the request, ap 3206 adjusts whether itwill allow orders to the ce 3216 for that auction. Only orders that areallowed to the ce 3216 are logged to disk and written to db 3208.However, any orders that are not accepted are logged (logd 3214) toprovide an audit trail.

13.7.2 Opening 3304

Opening an auction consists of the following steps:

-   -   ap 3206 receives the request to open from a uip 3202.    -   ap 3206 writes the request as a transaction to disk and to db        3208.    -   Updates the current state in db 3208.    -   uip's 3202 begin allowing users to make LOB requests based on        the updated state in db 3208.    -   ap 3206 begins allowing customer order requests from the uip's        3202 on that auction.    -   ap 3206 replies to the uip 3202 that requested the state change.

13.7.3 Closing 3302

Closing an auction consists of the following steps:

-   -   ap 3206 receives the request to close from a uip 3202.    -   ap 3206 writes the request as a transaction to disk and to db        3208.    -   ap 3206 updates the current state in db 3208.    -   uip's 3202 stop allowing users to make LOB requests based on the        updated state in db 3208.    -   ap 3206 stops allowing customer order requests from the uip's        3202 on that auction.    -   ap 3206 replies to the uip 3202 that requested the state change.

13.7.4 Finalization 3306

Finalizing an auction consists of the following steps:

-   -   ap 3206 receives the request to finalize from a uip 3202.    -   ap 3206 writes the finalization request as a transaction to disk        and to db 3208.    -   ap 3206 updates the current state in db 3208.    -   ap 3206 stops allowing any orders.    -   ap 3206 waits for all other transactions against this auction to        complete (ce 3216 will reply to the ap's 3206 last notification        of new transaction message). ap 3206 does not send a new        transaction message to the ce 3216 for the finalization        transaction.    -   ap 3206 reads all reports for this auction from db 3208 and        writes them in the last transaction for this auction.    -   ap 3206 replies to the uip 3202 that requested the state change.

13.7.5 Cancel 3308

Canceling an auction consists of the following steps:

-   -   ap 3206 kills the ce 3216 process for that auction.    -   ap 3206 writes the cancellation request as a transaction to disk        and to db 3208.    -   ap 3206 updates the current state in db 3208.    -   The auction effectively vanishes from the user interface, since        uip's 3202 will no longer accept any requests for that auction.    -   ap 3206 replies to the uip 3202 that requested the state change.

13.7.6 Opening Orders

Opening orders will be entered on the vanilla replicating claims, whichare described in section 11.2.2 and 11.3.1. These orders are constructedby the system using the total market making capital defined in theauction which is proportionately spread across all opening orders usingthe opening prices.

13.7.7 Customer Fees

Customers may be charged fees based on their total cleared replicatedpremium. Fees may vary depending on instrument type using the parametersdigitalFee, vanillaFee, digitalComboFee, vanillaComboFee, or forwardFee.

Each of these fees specifies the amount that the customer is charged perfilled contract on an order. This fee will typically be added to thefinal premium that the customer either must pay or receive for an order.The fee may also optionally be used to adjust the final price for theorder.

13.8 Startup

Tasks must be started (to completion) in roughly the following order:

-   -   1. resd 3204/db 3208/logd 3214    -   2. dp 3210    -   3. ap 3206/lp 3212    -   4. uip 3202

Information from disk must be loaded in the following order:

-   -   1. desk    -   2. groups    -   3. users    -   4. events    -   5. auctions

13.8.1 Loading Events at Startup

Events are loaded into db 3208 at startup by the ap 3206. Sincemodifications are not allowed, once they go in they remain constant forthe session.

13.8.2 Loading Auctions at Startup

Auctions are loaded at startup from their disk directory into db 3208 byap 3206. This is done by sequentially reading the transaction files andwriting them to db 3208. When complete, a ce 3216 is started and sent anotification of new transactions for the auction.

13.8.3 Loading Desk/Users/Groups at Startup

The desk, users and groups are loaded into db 3208 from their diskdirectories at startup by dp 3210. This is done by sequentially readingthe transaction files and writing them to db 3208.

13.9CE 3216 Implementation 13.9.1 CE Implementation PerformanceOptimization and Benefits

The performance of the ce 3216 is the major factor which enables usersto interact with the system and receive accurate real-time indicationsof current prices and fills as well as limit order book calculations.This is accomplished by minimizing the time lag between when an order isreceived and its effect is reflected in the prices and order fills thatare reported to users through the user interface. This capabilitysignificantly differentiates this system implementation from auctionbased systems that do not provide feedback to users on pricing, orderfills and “what-if” scenarios (limit order book) in real-time.

There are a number of optimizations contained in this implementation,which contribute to minimizing the amount of time required to calculatean equilibrium. In practice the combination of these optimizations hasresulted in performance improvements of up to 3 orders of magnitude overimplementations which do not use these techniques, across a wide varietyof realistic test cases. The use of these optimization techniques then,makes a fundamental difference in the behavior and usage of the systemfrom the users perspective. Specifically, the following techniques areused to effect this optimization in speed:

-   -   ce's 3216 and le's 3218 are run as separate processes on        separate processors and do not contend for system CPU resources.    -   Each ce 3216 utilizes a dedicated processor so there is no        contention for CPU resources between ce's 3216 running different        simultaneous auctions.    -   Orders for a ce 3216 are aggregated in internal data structures        when orders are at the same limit prices, strike, and option        type. This reduces the amount of data elements that the ce 3216        has to loop over when computing fills.    -   In updatePrices 3506, only the non-zero elements of the        replication weight vectors are processed and all orders on the        same option are aggregated since they are all executed at the        same price regardless of their limit prices.    -   The accelerate function is used in convergePrices 3510 to        accelerate the stepping under certain conditions, greatly        increasing the convergence speed.    -   The opening orders (section 13.8.14) are scaled appropriately        greatly increasing the speed of convergence when the amounts of        opening orders are large relative to the amount of premium in        the system. This condition commonly occurs at the beginning of        an auction.    -   Equilibriums are calculated using hot start method (described in        convergePrices in section 13.9.3) coupled with the use of        phaseTwo 3516. If hot start is used without this step, it is        much faster than cold start, but the prices and fills are        inconsistent. If cold start is used, the prices and fills are        consistent, but the system is much slower. Hot start combined        with phaseTwo 3516 yields much faster computation without any        pricing or fill inconsistencies.    -   Only the orders that have limit prices within priceGran of the        respective market price are inputted to the lp in runLp 3518,        even though the obvious approach would be to send all of the        orders. This approach significantly reduces the size of the lp        problem and reduces the time to compute the results, without        affecting the quality.    -   The memory for the replication weights is allocated on a 16 byte        boundary. This is to take advantage of the Pentium 4 SIMD        architecture so that the compiler can optimize dot products.        This is important in calculating the price for an option        (updatePrices 3506) and in updating the fill of an option        (setFill 4202).    -   The approach of stepping all order fills at once (see FIG. 35        and section 7.9) or “vector stepping” and then computing the        equilibrium provides significant speed improvements over        computing an equilibrium after any fill is stepped.    -   The method for solving rootFind 3504 based on Newton-Raphson        provides significant speed improvements over other numerical        solution techniques.

13.9.2 EqEngine (Equilibrium Engine) Object

The EqEngine object encapsulates the core equilibrium algorithm and thestate (data) associated with running it for a single auction. It is usedto implement both the ce 3216 and le 3218.

Its methods are:

Method Description InitEqEngine 3404 The InitEqEngine 3404 functionshown in FIG. 58 initializes the data in the EqEngine. RunEqEngine 3406The RunEqEngine 3406 function, as shown in FIG. 35, is the mainexecutive function for an equilibrium calculation AddTxToEqEngine 3408The addTxToEqEngine 3408 function shown in FIG. 54 takes an order andmakes adjustments to its limit price and replication weights. The limitprice is adjusted by calling the adjustLimitPrice method for the optiontype of the order. The replication weights are determined by determiningif the order is a buy or a sell. If the order is a sell, the sellPayoutsare selected which are the replication weights for the complementarybuy.Its data structures are defined as follows:

13.9.2.1 OptionDef Object

There is an optionDef object for each unique traded instrument within anauction. At the creation of the object, the computePayouts 5900 methodshown in FIG. 59 is called, which generates the buyPayouts[ ],sellPayouts[ ] and negA[ ] vectors. These vectors only need to begenerated once and stay persistent throughout the lifetime of theauction. A new optionDef object is created when a new unique customerinstrument is requested. This object encapsulates the following data:

Data Description optionType The option type, e.g., vanilla call, digitalrisk reversal. strike The option's lower strike value. spread Theoption's higher strike value. buyPayouts[numRepClaims] The replicationweight vector for a buy of this option. This vector is passed to theEqEngine as part of the trade object for a buy of this option. Thisvector is computed using equations 11.2.3A-11.2.3D. Each vector elementis denoted as a_(s), s = 1, 2, . . . , 2S − 2. sellPayouts[numRepClaims]The replication weight vector for a sell of this option (sell replicatedas a complementary buy). This vector is passed to the EqEngine as partof the trade object for a sell of this option. This vector is computedusing equations 11.2.3E-11.2.3H. Each vector element is denoted asa_(s), s = 1, 2, . . . , 2S − 2. negA[numRepClaims] The weights vectorused to calculate the price of the option. price The price of the optiondenoted by π_(j) in section 11. priceGran The tolerance to converge theprice for this option priceAdjust The market price is calculated fromthe replication price by subtracting priceAdjust.This object encapsulates the following methods:

Method Description computePayouts Initializes the buyPayouts[ ],sellPayouts and negA[ ] arrays in the optionDef. adjustLimit Convertsthe customer order limit price to the replicated limit price for theEqEngine. Described in section 11.4.4 and denoted by w_(j) ^(a).updatePrice Computes the market price of the option given the vanillareplication claim prices and denoted by π_(j) in section 11.

13.9.2.2 Global Variables

The global variables are as follows:

Global Variables Description numRepClaims Number of replicating claimsin the vanilla replicating basis. This quantity equals 2S − 2 in section11.2.2. numOptions The number of options with unique replication weightvectors. This quantity is less than or equal to J based on the notationsection 11. optionList[numOptions] Array of option objects.openPremium[numRepClaims] Opening order premium for the vanillareplicating claims. This vector has sth element θ_(s) in section 11, s =1, 2, . . . , 2S − 2. notional[numRepClaims] Aggregated filled notionalfor the vanilla replicating claims. This vector has sth element y_(s) insection 11, s = 1, 2, . . . , 2S − 2. price[numRepClaims] Price of thevanilla replicating claims. These quantities are denoted by p_(s) insection 11. totalInvested Total replicated cleared premium. Thisquantity denoted by M in section 11.

13.9.2.3 Constants

The constants are as follows:

Constant Description ACCEL_LOOP The number of iterations before callingaccelerate 3604 function. This value was empirically chosen to be 60.STEP_LOOP The number of iterations before adjusting the step size. Thisvalue was empirically chosen to be 6. CON_LOOP The number of iterationsbefore checking for convergence. This value was empirically chosen to be96. MIN_K The minimum average opening premium amount. This value is1000. Used in scaling function. MAX_ITER The maximum number ofiterations for Newton Raphson convergence. It is set to 30. INIT_STEPThe initial step size for an order. It is set to 0.1. GAMMA_PT If theratio of bigNorm to smallNorm is above GAMMA_PT then the step size isincreased else decrease step size. It is set to 0.6. ALPHA This is anaveraging constant used in step size selection. It is set to 0.25.MIN_STEP_SIZE The minimum permitted step size. It is set to 1e−9.ALPHA_FILL This is an acceleration averaging constant. It is set to 0.8.ACCEL This constant is used in acceleration. It is set to 7. PRICE_THRESThis constant is the tolerance that the vanilla replication prices mayvary by in the lp. It is set to 1e−9. ROUND_UP This is used in roundingthe vanilla replication prices. It is set to 1e−5.

13.9.2.4 Trade Object

The trade object encapsulates the following data:

Data Description requested The amount requested for the order denoted asr_(j) in section 11. limit The adjusted limit price for the EqEnginedenoted as w_(j) ^(a) in section 11. priceGran The tolerance to convergethe pricing to for this order. A[numRepClaims] The replication weightsfor the option. This vector has sth element a_(s) in section 11 and iscomputed using equations 11.2.3A-11.2.3H.

13.9.2.5 Option Object

The option objects encapsulates the following data:

Data Description price The price of the option. This quantity is denotedwith the variable π_(j) in section 11. A[numRepClaims] The replicationweights for the option. These quantities denoted by a_(s) in section 11and are computed using equations 11.2.3A-11.2.3H. orders Pointer to thehead of the orders linked list numOrders Length of linked list.priceGran Tolerance to converge prices within limit price currentPointer to the current active order in linked list activeHead Pointer toorder with highest limit price for phase 2 convergence

13.9.2.6 Order Object

The order objects encapsulates the following data:

Data Description head Pointer to the next order with lower limit price.tail Pointer to the next order with higher limit price. limit Limitprice of order. This quantity is denoted by w_(j) in section 11.requested The requested amount before scaling. This quantity is denotedby r_(j) in section 11. invest The scaled requested amount. filled Theamount filled. This quantity is denoted by x_(j) in section 11.smallNorms Used in step size selection.. bigNorms Used in step sizeselection. step Current step size. gamma Used in step size selection.runFilled Used by step size acceleration function. lastFill Used by stepsize acceleration function. active Indicates this order is included byphaseTwo stepping.

13.9.2.7 ce Report

The ce report encapsulates the following data:

Data Description totalInvested Total replicated cleared premium. Denotedas M in section 11. numRepClaims Number of replicating claims in thevanilla replicating basis. Equal to 2S-2 in section 11.2.2.openPremium[numRepClaims] Opening order premium for the vanillareplicating claims. Vector of length 2S-2 with sth element θ_(s), usingthe notation in section 11. notional[numRepClaims] Aggregated fillednotional for the vanilla replicating claims. Vector of length 2S-2 withsth element y_(s) using the notation from section 11. prob[numRepClaims]Price of the vanilla replicating claims. Vector of length 2S-2 with sthelement p_(s) using the notation from section 11. numOptions The numberof options with unique replication weight vectors.aList[numOptions][numRepClaims] The replication weights for each option.granList[numOptions] Tolerance to converge prices within limit price foreach option priceList[numOptions] The price of each option denoted byπ_(j) in section 11. numTrades The number of trades with uniquereplication weight and limit prices opIdxList[numTrades] The index intooptionList[ ] corresponding to the option for each trade.limitList[numTrades] The limit price of each trade denoted by w_(j) insection 11. amountList[numTrades] The requested amount for each tradedenoted by r_(j) in section 11. fillList[numTrades] The amount filledfor each trade denoted by x_(j) in section 11.

13.9.3 ce 3216 Top Level Processing

The top-level processing of the ce 3216 is shown in FIG. 34. Theapplication architecture shows that the ap 3206 process stores auctiontransactions (tx) in db 3208. There are three kinds of auctiontransactions: configuration transactions, state change transactions andorder transactions. When ce 3216 is started, it reads all auctiontransactions from db 3208. State change transactions are ignored, andthe last configuration transaction is noted. All order transactions arekept in txList. After reading all available transactions from db 3208for this auction, the most recent (last) configuration transaction isstored in db 3208 as the current auction definition. Then the ordertransactions in txList are processed and used to create the updatedorder/fills report and ce report which are stored in the db 3208. Theprices for each order are determined by reading the optionDef.priceelement for the orders' option type. The prices in all optionDef objectsare updated by using the vanilla replicating claims prices in theEqEngine price[ ] array. Each order fill is read from the EqEngine'sorder objects. If orders have been aggregated in the EqEngine then thefills are split pro-rata among the aggregated orders.

After processing of the initial order set is finished, ce 3216 waits fornotification from ap 3206, upon which it reads new transactions out ofdb 3208. If a configuration transaction is seen, ce 3216 terminates andthen is immediately restarted by resd 3204. This allows theconfiguration transaction to change data such as strikes, which are partof the initialization of the ce.

The following sections describe the implementation of RunEq Engine, thecore algorithm.

13.9.4 convergePrices 3510

The convergePrices 3510 function shown in FIG. 36 increases or decreasesthe fills on orders until the constraints in equation 11.4.6A are met.This equilibrium is calculated using the last equilibrium prices andfills as the initial starting point. This is referred to as “hotstarting” the equilibrium calculation, which is significantly fasterthan resetting all the order fills to 0 and recalculating theequilibrium each time new orders are processed (cold start). Theconvergence algorithm is discussed in detail in section 7.

13.9.5 updatePrices 3506

The updatePrices 3506 function shown in FIG. 37 iterates through all theoptions defined in the optionList[ ] and calculates the price for eachoption. The price of an option is the dot product of the replicatingvanilla claims prices vector (price[ ]) and the option replicationweight vector (option.A[ ]), as described in equation 11.4.3C. Since allorders on the same option have the same price there is no need tocalculate the price for every order. The replication weight vectors alsohave many that are equal to zero. The code is optimized by looping overthe start and end indices of the non-zero elements.

13.9.6 rootFind 3504

The rootFind 3504 function shown in FIG. 38 computes the totalinvestment amount (totalinvested, denoted as M in section 11 and as T insection 7) and the prices of the vanilla replication claims. Thisfunction uses the Newton-Raphson algorithm to solve Equation 7.4.1(b).The input to this function is the aggregated filled notional amounts(denoted as the y_(s)'s in section 7 and 11) for the replicating vanillaclaims (notional[ ]).

13.9.7 initialStep 3508

The initialStep 3508 function shown in FIG. 39 initializes the variablesused to control order stepping for all orders. This function initializesthe variables used in section 7.9.

13.9.8 stepOrders 3602

The stepOrders 3602 function shown in FIG. 40 iterates through alloptions in the optionList[ ]. The option.current pointer identifies theorder in the orders linked list that is partially filled. All ordersabove this order in the list have higher limit prices and are fullyfilled; and all orders below this order have lower limit prices and have0 fill. If the price is above order.limit then remove order.step fromorder.filled else add order.step to order.filled. If an order becomesfully filled or zero filled then the next current order is selected bysearching the list in the appropriate direction. This functionimplements the logic from section 7.9, specifically in step 8.

13.9.9 selectStep 3608

The selectStep 3608 function shown in FIG. 41 is called at everySTEP_LOOP iteration to adjust each order's step size. The step size isadjusted based on the absolute change in fill versus the absolute sum ofthe steps made. This ratio is between 0 and 1. If the ratio is aboveGAMMA_PT (0.6), the step size is increased; if it is below the step sizeis reduced. This function implements the dynamic step size approach fromsection 7.9, specifically in step 8(b).

13.9.10 Accelerate 3604

The accelerate 3604 function shown in FIG. 42 integrates the progressmade by each order in the previous ACCEL_LOOP iterations and adjusts thefill on the order. This function greatly improves convergence speed.

13.9.11 setFill 4202

The setFill 4202 function shown in FIG. 43 takes the fill and the orderas an input and updates the vanilla replication notional. It implementsthe dot product of replication weights with the fill as described inequation 11.4.5B.

13.9.12 checkConverge 3606

The checkConverge 3606 function shown in FIG. 44 checks if the algorithmhas converged using the approach from section 7.9, specifically in step8(a). This function verifies that the following conditions are met forall orders.

-   -   1. If order.price>(order.limitPrice+order.priceGran) then        order.filled=order.requested    -   2. If order.price<(order.limitPrice−order.priceGran) then        order.filled=0    -   3. If neither of the above conditions is met then        0≦order.filled≦order.requested

13.9.13 addFill 4002

The addFill 4002 function shown in FIG. 45 increases the fill on anorder by step size and updates step size variables.

13.9.14 decreaseFill 4004

The decreaseFill 4004 function shown in FIG. 46 decreases the fill on anorder by step size and updates the step size variables.

13.9.15 scaleOrders 3502

If the average opening order premium amount (denoted by

$\sum\limits_{s = 1}^{{2S} - 2}{\theta_{s}/\left( {{2S} - 2} \right)}$

in section 11) is greater than MIN_K then all order requested amountsand opening order premium are scaled down.

${{The}\mspace{14mu} {scale}\mspace{14mu} {factor}} = {\sum\limits_{s = 1}^{{2S} - 2}{{\theta_{s}/\left( {\left( {{2S} - 2} \right)*{MIN\_ K}} \right)}{\left. \quad \right).}}}$

If the scale factor is <1 then it is set to 1. This technique,scaleOrders 3502 shown in FIG. 47, reduces the maximum range that thestepping algorithm has to cover to fill an order. This approach speedsup convergence for auctions with large opening order amounts.

13.9.16 phaseTwo 3516

The prices and fills may be slightly different between a hot started anda cold started equilibrium calculation. The second convergence phase,phaseTwo 3516 shown in FIG. 48 rectifies the problem of fills and priceschanging when orders worse than the market are added. The approach is asfollows:

-   -   1. Calculate an equilibrium using hot start.    -   2. Round the vanilla replicating claim prices to eliminate        noise.    -   3. Identify orders within the tolerance 2*priceGran of their        limit prices.    -   4. Set the fills for these orders to 0 and reset the stepping        variables to their initial values.    -   5. Recalculate the equilibrium by only stepping the orders        identified in step 3.        The initial conditions for phase two convergence will be the        same for hot started equilibriums even if orders worse than the        market are added. The tolerance 2*priceGran is chosen because        hot starting the algorithm may result in prices being off by        priceGran tolerance.

13.9.17 runLp 3518

runLp 3518 shown in FIG. 49 uses a linear program code to maximize thecleared premium in the auction. After an equilibrium has been reachedthere are three scenarios for an order:

-   -   1. Order is fully filled and its limit price is greater than        priceGran above price.    -   2. Order has 0 fill and its limit price is less than priceGran        below price.    -   3. Order price is within +/− priceGran of its limit price.        Only the orders from case 2 are inputted to the lp 3212 as        variables which reduce the size of the lp 3212 problem. This        function implements the linear program discussed in step 8(c) of        section 7.9. This function uses the third party IMSL linear        program subroutine “imsl_d_lin_prog” produced by Visual Numerics        Inc.

13.9.18 roundPrices 3512

roundPrices 3512 shown in FIG. 50 rounds and normalizes the vanillareplication prices.

13.9.19 findActiveOrders 3514

The function findActiveOrders 3514 shown in FIG. 51 marks orders whoseprice is within a tolerance of their limit price as active. These orderswill be stepped in phase two convergence.

13.9.20 activeSelectStep 4804

The activeSelectStep 4804 function shown in FIG. 52 only adjusts thestep size on the active orders for phase two convergence.

13.9.21 stepActiveOrders 4802

The stepActiveOrders 4802 function shown in FIG. 53 only steps theactive orders for phase two convergence.

13.10 LE 3218 Implementation

As shown in FIG. 62-66, LOB requests are made by a uip 3202 to lp 3212,that forwards the request to an available le 3218. The le 3218 computeshow much volume is available at a series of limit prices above (forbuys) and below (for sells) the current indicative mid-market price fora particular option. It works by placing a very large order at a seriesof limit prices above the current price. The amount of this order,indicated by LOB_PROBE_AMOUNT in the figure, should be much larger thanthe orders in the equilibrium. This implementation uses1,000,000,000,000 (one trillion). The le 3218 uses all the datastructures and functions of ce 3216, but runs as a completely separateinstance of the EqEngine, and has no interaction with ce 3216. When aLOB request comes in, the le 3218 retrieves the latest ce report fromthe db 3208 and instantiates an EqEngine using this report that is inthe exact same state as the EqEngine from the ce 3216 that was used tocreate the ce report. The LOB request itself consists of these fields:

-   -   buyPayouts[ ]: Vector of per-state payouts.    -   sellPayouts[ ]: Vector of per-state payouts for complementary        order.    -   offsetList[ ]: Vector of offsets above/below price at which to        compute LOB.    -   lobGran: LOB granularity (the smallest increment between LOB        limit prices).    -   priceAdjust: For options that may have a negative price        (risk-reversals, forwards), this is used to scale limit prices        when entering orders into the EqEngine and to scale them back        when reporting results.        The offsets in request.offsetList[ ] are usually interpreted as        percentages above/below the mid-market price. If 1% of the        mid-market price is smaller than lobGran, the offsets are        interpreted as multiples of lobGran.

If two orders are placed on the same option, the order with the higherlimit price takes precedence. Therefore, to compute the LOB at a seriesof points above the market, we do not have to cancel each order inbetween calculations; it suffices to add each order one at a time, runthe equilibrium and read the volume out of the EqEngine. After computingall of the buy points, we cancel each of the buy orders and then move onto calculate the sell side of the LOB.

13.11 Network Architecture

As shown in FIG. 67, the network architecture provides an efficient andredundant environment for the operation of DBAR auctions.

13.11.1 Architectural Elements

The architectural elements are defined as follows:

Element Description PDC 6712 Primary Data center shown in FIG. 67 - theprimary location for hosting servers. The data center provides a securelocation with reliable/redundant power and internet connections. BDC6714 Backup Data center shown in FIG. 67 - the backup location forhosting servers. It is to be located sufficiently far from the PDC 6712so as not to be affected by the same power outages, natural disasters,or other failures. The data center provides a secure location withreliable/redundant power and internet connections. NOC 6708 NetworkOperations center shown in FIG. 67 - the location used to host theservers and staff that operate the system. CPOD 6724 Client pod shown inFIG. 67 - the group of servers and networking devices used to support aclient session at a data center. MPOD Management pod shown in FIG. 67 -the group of servers and 6718, 6720 network devices used to monitor andmanage the CPODs 6724 at a data center. The MPOD supports the followingfunctions: snmp monitoring of hardware in the data center collectssyslod eents from all devices in the data center runs applicationmonitoring tools hosts an authentication server which provides twofactor authentications for system administrators who access any serversor network devices. APOD 6722 Access Pod shown in FIG. 67 - the group ofnetwork devices that provides centralized, firewalled access to thepublic internet for a group of CPODs 6724 at a data center.

13.11.2 Devices

The devices are defined as follows:

Device type Description Typical Hardware ts 3222 Transaction servershown in FIG. 32 - 2 processor pentium-4 class runs the followingprocesses: PC server db 3208 ap 3206 dp 3210 lp 3212 resd 3204 logd 3214ws 3220 Web server shown in FIG. 32 - runs the 2 processor pentium-4class following processes: PC server uip 3202 cs 3224 Calculation servershown in FIG. 32 - 2 processor pentium-4 class runs the followingprocesses: PC server ce ls 3226 LOB server shown in FIG. 32 - runs the 2processor pentium-4 class following processes: PC server le 3218 msManagement server - runs system 2 processor pentium-4 class managementtools. PC server sw 6702 Switch shown in FIG. 67 - provides Cisco 3550100B/T switched ethernet connectivity tr 6816 Terminal server - provideaccess to Cisco 2511 console ports on all devices over ethernet gw 6704Gateway router shown in FIG. 67 - Cisco 2651 provides access to theinternet. fw 6706 Firewall shown in FIG. 67 - blocks all Cisco PIXinbound and outbound access except for port 80 and 443 (http and https).Performs stateful inspection of all packets.

13.11.3 CPOD 6724 Details

Each CPOD 6724 is used to host the software required to run auctions fora sponsor. A CPOD 6724 typically consists of the following:

Server Type Quantity Comments ts 3222 2 Redundant pair. ws 3220 4 Loadbalanced. cs 3224 2 Pool for active auctions - more servers will beadded as the requirement for more simultaneous auctions increases -typically allocate 1 processor per active auction. ls 3226 4 Pool foractive auctions - more servers may be added to reduce LOB response timeunder load. ms 2 Redundant pair.Since there are multiple CPODs 6724 at the PDC 6712, multiple sponsorscan run auctions simultaneously. As shown in FIG. 68, each individualCPOD 6724 in the PDC 6712 has a corresponding CPOD 6724 in the BDC 6714which is available for failover in the event of a major failure at thePDC 6712, such as a loss of power or connectivity.

13.12 FIGS. 32-68 Legend

Appendix 13A

The elements are defined as follows:

Element Name Description abstract A short text description of theauction. accessPrivileges Controls which screens and reports a user isallowed to access. The possible values and their meanings are: B -customers - can view, place, and modify orders C - administrators - sameas B, plus can create and modify auction details and state, can createevents, can create, modify, and delete users accessStatus Reflects thecurrent status of a user. The values and their meanings are: enabled -the user is allowed access. expired - the user's account has expired(see accountExpires) and will be denied access until a useradministrator changes the accessStatus. locked - the user's account hasbeen locked by the system due to a security violation and will be deniedaccess until a user administrator changes the accessStatus. disabled -the user's account has been disabled by the user administrator and willbe denied access until a user administrator changes the accessStatus.accountExpires The date that the user's account will expire. When thisdate is reached, the accountStatus will be set to expired. amount Theamount of an order. Depending on the optionType for a given order thismay have several meanings such as: For optionType = digitalPut,digitalCall, digitalRange, digitalStrangle, or digitalRiskReversal, theorder amount is the notional amount requested by the order. ForoptionType = vanillaFlooredPut, vanillaCappedCall, vanillaPutSpread,vanillaCallSpread, vanillaStraddle, vanillaStrangle,vanillaRiskReversal, or forward the order amount is the number ofoptions contracts requested by the order. Order amount is denoted byr_(j) for customer order j in section 11. auctionId The unique ID thesystem assigns to an auction when it is created. auctionSymbol A uniquesymbol for the auction. This symbol may be re-used after the deletion ofthe auction. canChangePsw Controls if a user is allowed to changehis/her own password. cap The cap (highest) strike used by the systemfor calculations in all auctions on a particular event. It is notvisible to the user and is denoted by k_(S−1) in section 11. currencyThe currency in which all auctions on a particular event aredenominated. It is a standard 3-letter ISO code. description An optionaltext field to describe the user. desk A unique name assigned byLongitude to identify a system configuration used by a sponsor.digitalComboFee The sponsor fee for digital strangle or risk reversaloptions in basis points of filled premium. digitalFee The sponsor feefor digital call, put or range options in basis points of filledpremium. email The email address of a user. end The date/time theauction ends. eventDescription A short text description of the event.eventId The unique ID the system assigns to an event when it is created.eventSymbol A unique symbol for the event. This symbol may not be reusedunless the event has been removed from the system. expiration The datethe options expire for a particular event. fill The current fill on anorder. firstName The first name of the user. floor The floor (lowest)strike used by the system for calculations in all auctions on aparticular event. It is not visible to the user and is denoted by k₁ insection 11. forwardFee The sponsor fee for forwards in basis points offilled premium. groupId The unique ID the system assigns to a group whenit is created. groupName The name of the group. At any given point,there is only one active (non-deleted) group for each groupName within asponsor, but there may be other groups with the same groupName that havebeen deleted previously. isCanceled This indicates if an order has beencanceled. isDeleted This indicates if the user or group has beendeleted. Note that users and groups are never actually deleted in thesystem but instead are simply marked as deleted. This is done topreserve referential integrity. lastName The last name of a user.limitOffsets The values used to specify the number of and location ofthe limit order book points for all auctions on a desk. limitPrice Thelimit price of an order. The limit price is denoted by w_(j) forcustomer order j in section 11. location The location of a user.lowerStrike The strike price for an option when optionType isdigitalCall, vanillaCappedCall, vanillaCall or vanillaStraddle. It isthe lower strike price for an option when optionType is digitalRange,digitalStrangle, digitalRiskReversal, vanillaCallSpread,vanillaPutSpread, vanillaStrangle, or vanillaRiskReversal.marketMakingCapital The capital supplied by the auction sponsor toinitially seed the equilibrium algorithm. mktPrice The current marketprice for an option. mustChangePsw This indicates that the user mustchange his password at the next login. openingPrices The initial pricesdisplayed by the system for an auction. optionType The type of option -the possible values are: digitalPut digitalCall digitalRangedigitalStrangle digitalRiskReversal vanillaPut vanillaFlooredPutvanillaCall vanillaCappedCall vanillaPutSpread vanillaCallSpreadvanillaStraddle vanillaStrangle vanillaRiskReversal forward A vanillaFlooredPut is a vanilla put spread whose lowest strike is the floor. AvanillCappedCall is a vanilla call spread whose highest strike is thecap. orderId The unique ID the system assigns to an order when it iscreated. payoutSettlement The payout settlement date of an auction.phone The phone number of a user. premiumCustomerPays The calculatedpremium amount that the customer must pay for a particular filled order.premiumCustomerReceives The calculated premium amount that the customerwill receive for a particular filled order. premiumSettlement Thepremium settlement date of an auction. price The pricing information foran option. pswChangedDate The date and time of the last time a user oradministrator changed a user's password. pswChangeInterval This is howoften (in days) a user must change his password. If zero, then thepassword does not have to be changed at a regular interval. revision Therevision of a desk, user, group, an auction, or an order. This starts at0, and increments by 1. revisionBy The userId of the person who made therevision. revisionDate The date and time of the revision. side Thisindicates if the order is a buy or a sell. sponsor The name of theauction sponsor. sponsorId The unique ID assigned by Longitude to anauction sponsor. It is used in users, groups and orders to identifytheir affiliation. start The starting date/time for an auction. This iscaptured for informational purposes only and is not enforced by thesystem. state The current state of the auction. The possible values are:open 3304 closed 3302 finalized 3306 canceled 3308 See the state diagramfor more information. The usage in of “state” in this section differsfrom the usage of the term state in section 11. strikes The set ofstrikes for an auction. Strikes are denoted by k₁, k₂, . . . , k_(S−1)in section 11. strikeUnits The units of the strikes for all auctions onan event. tickSize The minimum amount by which the underlying on anevent.can change denoted by ρ in section 11. tickValue The payout valueof a tick on an event.for a vanilla option. title A brief textdescription of an auction. upperStrike The strike price for an optionwhen optionType is digitalPut, vanillaFlooredPut or vanillaPut. It isthe upper strike price for an option when optionType is digitalRange,digitalStrangle, digitalRiskReversal, vanillaCallSpread,vanillaPutSpread, vanillaStrangle, or vanillaRiskReversal. userId Theunique ID the system assigns to a user when it is created. userName Theunique name used by a user to log in to the system. At any given point,there is only one active (non-deleted) user for each userName within asponsor, but there may be other users with the same userName that havebeen deleted previously. vanillaComboFee The sponsor fee for vanillastraddle, strangle or risk reversal options in basis points of filledpremium. vanillaFee The sponsor fee for vanilla call, put or spreadoptions in basis points of filled premium. vanillaPricePrecisionSmallest displayed precision for vanilla prices.

14. ADVANTAGES OF PREFERRED EMBODIMENTS

This specification sets forth principles, methods, and systems thatprovide trading and investment in groups of DBAR contingent claims, andthe establishment and operation of markets and exchanges for suchclaims. Advantages of the present invention as it applies to the tradingand investment in derivatives and other contingent claims include:

-   (1) Increased liquidity: Groups of DBAR contingent claims and    exchanges for investing in them according to the present invention    offer increased liquidity for the following reasons:    -   (a) Reduced dynamic hedging by market makers. In preferred        embodiments, an exchange or market maker for contingent claims        does not need to hedge in the market. In such embodiments, all        that is required for a well-functioning contingent claims market        is a set of observable underlying real-world events reflecting        sources of financial or economic risk. For example, the quantity        of any given financial product available at any given price can        be irrelevant in a system of the present invention.    -   (b) Reduced order crossing. Traditional and electronic exchanges        typically employ sophisticated algorithms for market and limit        order book bid/offer crossing. In preferred embodiments of the        present invention, there are no bids and offers to cross. A        trader who desires to “unwind” an investment will instead make a        complementary investment, thereby hedging his exposure.    -   (c) No permanent liquidity charge: In the DBAR market, only the        final returns are used to compute payouts. Liquidity variations        and the vagaries of execution in the traditional markets do not,        in preferred embodiments, impose a permanent tax or toll as they        typically do in traditional markets. In any event, in preferred        embodiments of the present invention, liquidity effects of        amounts invested in groups of DBAR claims are readily calculable        and available to all traders. Such information is not readily        available in traditional markets.-   (2) Reduced credit risk: In preferred embodiments of the present    invention, the exchange or dealer has greatly increased assurance of    recovering its transaction fee. It therefore has reduced exposure to    market risk. In preferred embodiments, the primary function of the    exchange is to redistribute returns to successful investments from    losses incurred by unsuccessful investments. By implication, traders    who use systems of the present invention can enjoy limited    liability, even for short positions, and a diversification of    counterparty credit risk.-   (3) Increased Scalability: The pricing methods in preferred    embodiments of systems and methods of the present invention for    investing in groups of DBAR contingent claims are not tied to the    physical quantity of underlying financial products available for    hedging. In preferred embodiments an exchange therefore can    accommodate a very large community of users at lower marginal costs.-   (4) Improved Information Aggregation: Markets and exchanges    according to the present invention provide mechanisms for efficient    aggregation of information related to investor demand, implied    probabilities of various outcomes, and price.-   (5) Increased Price Transparency: Preferred embodiments of systems    and methods of the present invention for investing in groups of DBAR    contingent claims determine returns as functions of amounts    invested. By contrast, prices in traditional derivatives markets are    customarily available for fixed quantities only and are typically    determined by complex interactions of supply/demand and overall    liquidity conditions. For example, in a preferred embodiment of a    canonical DRF for a group of DBAR contingent claims of the present    invention, returns for a particular defined state are allocated    based on a function of the ratio of the total amount invested across    the distribution of states to the amount on the particular state.-   (6) Reduced settlement or clearing costs: In preferred embodiments    of systems and methods for investing in groups of DBAR contingent    claims, an exchange need not, and typically will not, have a need to    transact in the underlying physical financial products on which a    group of DBAR contingent claims may be based. Securities and    derivatives in those products need not be transferred, pledged, or    otherwise assigned for value by the exchange, so that, in preferred    embodiments, it does not need the infrastructure which is typically    required for these back office activities.-   (7) Reduced hedging costs: In traditional derivatives markets,    market makers continually adjust their portfolio of risk exposures    in order to mitigate risks of bankruptcy and to maximize expected    profit. Portfolio adjustments, or dynamic hedges, however, are    usually very costly. In preferred embodiments of systems and methods    for investing in groups of DBAR contingent claims, unsuccessful    investments hedge the successful investments. As a consequence, in    such preferred embodiments, the need for an exchange or market maker    to hedge is greatly reduced, if not eliminated.-   (8) Reduced model risk: In traditional markets, derivatives dealers    often add “model insurance” to the prices they quote to customers to    protect against unhedgable deviations from prices otherwise    indicated by valuation models. In the present invention, the price    of an investment in a defined state derives directly from the    expectations of other traders as to the expected distribution of    market returns. As a result, in such embodiments, sophisticated    derivative valuation models are not essential. Transaction costs are    thereby lowered due to the increased price transparency and    tractability offered by the systems and methods of the present    invention.-   (9) Reduced event risk: In preferred embodiments of systems and    methods of the present invention for investing in groups of DBAR    contingent claims, trader expectations are solicited over an entire    distribution of future event outcomes. In such embodiments,    expectations of market crashes, for example, are directly observable    from indicated returns, which transparently reveal trader    expectations for an entire distributions of future event outcomes.    Additionally, in such embodiments, a market maker or exchange bears    greatly reduced market crash or “gap” risk, and the costs of    derivatives need not reflect an insurance premium for discontinuous    market events.-   (10) Generation of Valuable Data: Traditional financial product    exchanges usually attach a proprietary interest in the real-time and    historical data that is generated as a by-product from trading    activity and market making. These data include, for example, price    and volume quotations at the bid and offer side of the market. In    traditional markets, price is a “sufficient statistic” for market    participants and this is the information that is most desired by    data subscribers. In preferred embodiments of systems and methods of    the present invention for investing in groups of DBAR contingent    claims, the scope of data generation may be greatly expanded to    include investor expectations of the entire distribution of possible    outcomes for respective future events on which a group of DBAR    contingent claims can be based. This type of information (e.g., did    the distribution at time t reflect traders' expectations of a market    crash which occurred at time t+1?) can be used to improve market    operation. Currently, this type of distributional information can be    derived only with great difficulty by collecting panels of option    price data at different strike prices for a given financial product,    using the methods originated in 1978 by the economists Litzenberger    and Breeden and other similar methods known to someone of skill in    the art. Investors and others must then perform difficult    calculations on these data to extract underlying distributions. In    preferred embodiments of the present invention, such distributions    are directly available.-   (11) Expanded Market for Contingent Claims: Another advantage of the    present invention is that it enables a well functioning market for    contingent claims. Such a market enables traders to hedge directly    against events that are not readily hedgable or insurable in    traditional markets, such as changes in mortgage payment indices,    changes in real estate valuation indices, and corporate earnings    announcements. A contingent claims market operating according to the    systems and methods of the present invention can in principle cover    all events of economic significance for which there exists a demand    for insurance or hedging.-   (12) Price Discovery: Another advantage of systems and methods of    the present invention for investing in groups of DBAR contingent    claims is the provision, in preferred embodiments, of a returns    adjustment mechanism (“price discovery”). In traditional capital    markets, a trader who takes a large position in relation to overall    liquidity often creates the risk to the market that price discovery    will break down in the event of a shock or liquidity crisis. For    example, during the fall of 1998, Long Term Capital Management    (LTCM) was unable to liquidate its inordinately large positions in    response to an external shock to the credit market, i.e., the    pending default of Russia on some of its debt obligations. This risk    to the system was externalized to not only the creditors of LTCM,    but also to others in the credit markets for whom liquid markets    disappeared. By contrast, in a preferred embodiment of a group of    DBAR contingent claims according to the present invention, LTCM's    own trades in a group of DBAR contingent claims would have lowered    the returns to the states invested in dramatically, thereby reducing    the incentive to make further large, and possibly destabilizing,    investments in those same states. Furthermore, an exchange for a    group of DBAR contingent claims according to the present invention    could still have operated, albeit at frequently adjusted returns,    even during, for example, the most acute phases of the 1998 Russian    bond crisis. For example, had a market in a DBAR range derivative    existed which elicited trader expectations on the distribution of    spreads between high-grade United States Treasury securities and    lower-grade debt instruments, LTCM could have “hedged” its own    speculative positions in the lower-grade instruments by making    investment in the DBAR range derivatives in which it would profit as    credit spreads widened. Of course, its positions by necessity would    have been sizable thereby driving the returns on its position    dramatically lower (i.e., effectively liquidating its existing    position at less favorable prices). Nevertheless, an exchange    according to preferred embodiments of the present invention could    have provided increased liquidity compared to that of the    traditional markets.-   (13) Improved Offers of Liquidity to the Market: As explained above,    in preferred embodiments of groups of DBAR contingent claims    according to the present invention, once an investment has been made    it can be offset by making an investment in proportion to the    prevailing traded amounts invested in the complement states and the    original invested state. By not allowing trades to be removed or    cancelled outright, preferred embodiments promote two advantages:    -   (1) reducing strategic behavior (“returns-jiggling”)    -   (2) increasing liquidity to the market    -   In other words, preferred embodiments of the present invention        reduce the ability of traders to make and withdraw large        investments merely to create false-signals to other participants        in the hopes of creating last-minute changes in closing returns.        Moreover, in preferred embodiments, the liquidity of the market        over the entire distribution of states is information readily        available to traders and such liquidity, in preferred        embodiments, may not be withdrawn during the trading periods.        Such preferred embodiments of the present invention thus provide        essentially binding commitments of liquidity to the market        guaranteed not to disappear.-   (14) Increased Liquidity Incentives: In preferred embodiments of the    systems and methods of the present invention for trading or    investing in groups of DBAR contingent claims, incentives are    created to provide liquidity over the distribution of states where    it is needed most. On average, in preferred embodiments, the implied    probabilities resulting from invested amounts in each defined state    should be related to the actual probabilities of the states, so    liquidity should be provided in proportion to the actual    probabilities of each state across the distribution. The traditional    markets do not have such ready self-equilibrating liquidity    mechanisms—e.g., far out-of-the-money options might have no    liquidity or might be excessively traded. In any event, traditional    markets do not generally provide the strong (analytical)    relationship between liquidity, prices, and probabilities so readily    available in trading in groups of DBAR contingent claims according    to the present invention.-   (15) Improved Self-Consistency: Traditional markets customarily have    “no-arbitrage” relationships such as put-call parity for options and    interest-rate parity for interest rates and currencies. These    relationships typically must (and do) hold to prevent risk-less    arbitrage and to provide consistency checks or benchmarks for    no-arbitrage pricing. In preferred embodiments of systems and    methods of the present invention for trading or investing in groups    of DBAR contingent claims, in addition to such “no-arbitrage”    relationships, the sum of the implied probabilities over the    distribution of defined states is known to all traders to equal    unity. Using the notation developed above, the following relations    hold for a group of DBAR contingent claims using a canonical DRF:

$\begin{matrix}{r_{i} = {\frac{\left( {1 - f} \right)*{\sum\limits_{i}T_{i}}}{T_{i}} - 1}} & (a) \\{q_{i} = {\frac{1 - f}{r_{i} + 1} = \frac{T_{i}}{\sum\limits_{i}T_{i}}}} & (b) \\{{\sum\limits_{i}q_{i}} = 1} & (c)\end{matrix}$

-   -   In other words, in a preferred embodiment, the sum across a        simple function of all implied probabilities is equal to the sum        of the amount traded for each defined state divided by the total        amount traded. In such an embodiment, this sum equals 1. This        internal consistency check has no salient equivalent in the        traditional markets where complex calculations are typically        required to be performed on illiquid options price data in order        to recover the implied probability distributions.

-   (16) Facilitated Marginal Returns Calculations: In preferred    embodiments of systems and methods of the present invention for    trading and investing in groups of DBAR contingent claims, marginal    returns may also be calculated readily. Marginal returns r^(m) are    those that prevail in any sub-period of a trading period, and can be    calculated as follows:

$\begin{matrix}{r_{i,{t - 1},t}^{m} = \frac{{r_{i,t}*T_{i,t}} - {r_{i,{t - 1}}*T_{i,{t - 1}}}}{T_{i,t} - T_{i,{t - 1}}}} & (1)\end{matrix}$

-   -   where the left hand side is the marginal returns for state i        between times t−1 and t; r_(i,t) and are the unit returns for        state i at times t, and t−1, and T_(i,t) and are the amounts        invested in state i at times t and t−1, respectively.        -   In preferred embodiments, the marginal returns can be            displayed to provide important information to traders and            others as to the adjustment throughout a trading period. In            systems and methods of the present invention, marginal            returns may be more variable (depending on the size of the            time increment among other factors) than the returns which            apply to the entire trading period.

-   (17) Reduced Influence By Market Makers: In preferred embodiments of    the systems and methods of the present invention, because returns    are driven by demand, the role of the supply side market maker is    reduced if not eliminated. A typical market maker in the traditional    markets (such as an NYSE specialist or a swaps book-runner)    typically has privileged access to information (e.g., the limit    order book) and potential conflicts of interest deriving from dual    roles as principal (i.e., proprietary trader) and market maker. In    preferred embodiments of the present invention, all traders have    greater information (e.g., investment amounts across entire    distribution of states) and there is no supply-side conflict of    interest.

-   (18) Increased Ability to Generate and Replicate Arbitrary Payout    Distributions: In preferred embodiments of the systems and methods    of the present invention for investing and trading in groups of DBAR    contingent claims, traders may generate their own desired    distributions of payouts, i.e., payouts can be customized very    readily by varying amounts invested across the distribution of    defined states. This is significant since groups of DBAR contingent    claims can be used to readily replicate payout distributions with    which traders are familiar from the traditional markets, such as    long stock positions, long and short futures positions, long options    straddle positions, etc. Importantly, as discussed above, in    preferred embodiments of the present invention, the payout    distributions corresponding to such positions can be effectively    replicated with minimal expense and difficulty by having a DBAR    contingent claim exchange perform multi-state allocations. For    example, as discussed in detail in Section 6 and with reference to    FIGS. 11-18, in preferred embodiments of the system and methods of    the present invention, payout distributions of investments in DBAR    contingent claims can be comparable to the payout distributions    expected by traders for purchases and sales of digital put and call    options in conventional derivatives markets. While the payout    distributions may be comparable, the systems and methods of the    present invention, unlike conventional markets, do not require the    presence of sellers of the options or the matching of buy and sell    orders.

-   (19) Rapid implementation: In various embodiments of the systems and    methods of the present invention for investing and trading in groups    of DBAR contingent claims, the new derivatives and risk management    products are processed identically to derivative instruments traded    in the over-the-counter (OTC) markets, regulated identically to    derivative instruments traded in the OTC markets and conform to    credit and compliance standards employed in OTC derivatives markets.    The product integrates with the practices, culture and operations of    existing capital and asset markets as well as lends itself to    customized applications and objectives.

In addition to the above advantages, the demand-based trading system mayalso provide the following benefits:

-   -   (1) Aggregation of liquidity: Fragmentation of liquidity, which        occurs when trading is spread across numerous strike prices, can        inhibit the development of an efficient options market. In a        demand-based market or auction, no fragmentation occurs because        all strikes fund each other. Interest in any strike provides        liquidity for all other strikes. Batching orders across time and        strike price into a demand-based limit order book is an        important feature of demand-based trading technology and is the        primary means of fostering additional liquidity.    -   (2) Limited liability: A unique feature of demand-based trading        products is their limited liability nature. Conventional options        offer limited liability for purchases only. Demand-based trading        digital options and other DBAR contingent claims have the        additional benefit of providing a known, finite liability to        option sellers, based on the notional amount of the option        traded. This will provide additional comfort for short sellers        and consequently will attract additional liquidity, especially        for out-of-the money options.    -   (3) Visibility/Transparency: Customers trading in demand-based        trading products can gain access to unprecedented transparency        when entering and viewing orders. Prices for demand-based        trading products (such as digital options) at each strike price        can be displayed at all times, along with the volume of orders        that would be cleared at the indicated price. A limit order book        displaying limit orders by strike can be accessible to all        customers. Finally, the probability distribution resulting from        all successful orders in the market or auction can be displayed        in a familiar histogram form, allowing market participants to        see the market's true consensus estimate for possible future        outcomes.        -   Demand-based trading solutions can use digital options,            which may have advantages for measuring market expectations:            the price of the digital option is simply the consensus            probability of the specific event occurring. Since the            interpretation of the pricing is direct, no model is            required and no ambiguity exists when determining market            expectations.    -   (4) Efficiency: Bid/Ask spreads in demand-based trading products        can be a fraction of those for options in traditional markets.        The cost-efficient nature of the demand-based trading mechanism        translates directly into increased liquidity available for        taking positions.    -   (5) Enhancing returns with superior forecasting: Managers with        superior expertise can benefit from insights, generating        significant incremental returns without exposure to market        volatility. Managers may find access to digital options and        other DBAR contingent claims useful for dampening the effect of        short-term volatility of their underlying portfolios.    -   (6) Arbitrage: Many capital market participants engage in        macroeconomic ‘arbitrage.’ Investors with skill in economic and        financial analysis can detect imbalances in different sectors of        the economy, or between the financial and real economies, and        exploit them using DBAR contingent claims, including, for        example, digital options, based on economic events, such as        changes in values of economic statistics.

15. ENHANCED PARIMUTUEL WAGERING

This section introduces example embodiments of enhanced parimutuelwagering, a method that increases the attractiveness of wagering onhorse races and other sporting events.

The outline for this section is as follows. Section 15.1 suggestsshortcomings with current wagering techniques and summarizes exampleembodiments of enhanced parimutuel wagering. Section 15.2 details themathematics of example embodiments. Section 15.3 illustrates enhancedparimutuel wagering with a detailed discussion of wagering on a threehorse race. Section 15.4 shows how example embodiments can be used inother settings.

15.1Background and Summary of Example Embodiments

This section provides background on different wagering techniques andsummarizes example embodiments. Section 15.1.1 introduces many of theterms that are used in this section. Section 15.1.2 describes parimutuelwagering, which is widely used for wagering on horse races in the U.S.and abroad. Section 15.1.3 suggests some shortcomings of parimutuelwagering. Section 15.1.4 introduces gaming against the house, which iswidely used in casinos for wagering on sporting events. Section 15.1.5discusses some shortcomings of this wagering technique. Section 15.1.6summarizes example embodiments of enhanced parimutuel wagering. Finally,section 15.1.7 describes the advantages of enhanced parimutuel wageringover parimutuel wagering and gaming against the house.

15.1.1Background and Terms

The term wagering association refers to a company that runs organizedand legal gaming, such as an authoriied casino, an authorized racingassociation (which runs wagering on horse or dog races, for instance),or a legal lottery organization. The wagering association determines afuture underlying event including, but not limited to, a horse race, asporting event, or a lottery. This underlying event must have multipleoutcomes that can be measured or otherwise objectively determined.During a pre-specified time period or so-called betting period, thewagering association allows bettors to make bets on the outcome of theunderlying event.

In making a bet, the bettor specifies an outcome or set of outcomes ofthe underlying event, and the bettor submits premium. If the specifiedoutcome(s) does not occur and the bet loses, then the bettor receives nopayout and loses the premium. If the specified outcome(s) occurs and thebet wins, then the wagering association pays the bettor an amount equalto the bet's payout amount. The bet's profit is the payout amount of thebet minus the premium amount of the bet. A bet's odds are the bet'sprofit per $1 of premium paid. For instance, if $10 in premium is bet,and the bettor receives a $50 pay out if the bet wins, then the bet hasa profit of $40 and the odds of the bet are 4 to 1.

15.1.2 Parimutuel Wagering

To illustrate parimutuel wagering on horse races, consider betting onthe winner of a horse race. During the betting period (which typicallytakes place in the time period leading up to the start of the horserace), the racing association accepts bets on which horse will win therace. When making such a win bet, the bettor specifies the horse to winthe race and submits the bet's premium amount. The racing associationtakes a fixed percentage (known as the track take) of the premium asrevenue for itself and puts the remaining money into the win pool. Oncethe race has begun (or “at the bell”), the racing association stopsaccepting bets for that race. After the horse race is over and thewinner has been determined, the racing association distributes theamount in the win pool to the bettors who bet on the winning horse inthe amount proportion to each winning bettor's premium.

In parimutuel wagering, the odds on a horse to win are determined by thetotal amount bet on each of the horses: the more that is bet on a horserelative to other horses in the race, the lower the odds and the lowerthe profit if the horse wins. In parimutuel wagering, all identical bets(e.g., a specific horse to win) have identical final odds, regardless ofthe time the bet is made during the betting period. This differs fromgaming against the house where the odds can vary over the betting period(as the casino adjusts the odds).

Assume, for example, that $100,000 is the total amount bet on horses towin the race and assume that the track take is 13%. Thus, the amount inthe win pool is equal to $87,000. Assume that $17,400 is bet on horse 1to win. If horse 1 wins, then the winning tickets totaling $17,400 willshare the $87,000 in the pool. If a bettor had bet $174 in premium onhorse 1 to win, then that bettor will be entitled to 1% of the win poolif horse 1 wins, where 1% equals $174 divided by $17,400. Therefore, abet of $174 in premium receives a payout of $870 if horse 1 wins, where$870 equals $87,000 multiplied by 1%. Thus, the bet's profit will be$696, where $696 equals $870 minus $174. Thus, the odds on horse 1 towin are 4 to 1, where 4 equals $696 divided by $174.

At the time a bet is made, the bettor does not know the odds and thepayout amount of the bet since the amount in the parimutuel pool is notfinal until after the start of the race. However, the racing associationprovides the bettor with indicative odds. The indicative odds aredetermined by the total amount bet on each of the horses up to thatpoint: the more that is bet on a horse relative to other horses in therace, the lower the indicative odds and the lower the indicative payoutif the horse wins. The indicative odds are not the final odds that thebettor receives on his/her bet. In fact, the final odds can besignificantly different than the indicative odds.

In the parimutuel system, the racing association's revenue on a singlerace with win bets is the track take multiplied by the amount of moneywagered. Thus, the racing association makes the same amount of moneyregardless of which horse wins the race, and the racing association hasno exposure or risk regarding the outcome of the horse races. (Thisdiscussion ignores the issue of breakage. See William Ziemba and DonaldHausch, Dr. Z's Beat the Racetrack, 1987, William Morrow for adiscussion of breakage.) The odds are determined mathematically bycomputer, and so the racing association does not require staff toconstantly update and quote odds and monitor the racing association'srisk in a horse race.

In addition to bets on horses to win, racing associations accept anumber of other types of bets, as displayed in Table 15.1.2A below.

TABLE 15.1.2A Types of bets that can be made on a horse race. Type ofBet Bet Pays Out if A place bet The bettor's horse finishes first orsecond A show bet The bettor's horse finishes first, second, or third Anexacta bet The bettor correctly selects the first place finisher and the(also called a second place finisher of the race in their exact perfectabet) finishing order A quinella bet The bettor correctly selects thefirst place finisher and the second place finisher without regard totheir finishing order

There is a separate parimutuel pool for each bet type, and all bets ofthe same type are entered into the same pool. For example, all win betsare entered into the win pool and all exacta bets are entered into theexacta pool. Thus, in current parimutuel wagering, the total amount inthe exacta pool and payouts from the exacta pool do not depend on theamount or relative amounts in the win pool.

15.1.3 Shortcomings of Parimutuel Wagering

The current technology for parimutuel systems has many shortcomings forthe bettors including the following.

-   -   Uncertainty of Payout. As mentioned above, the bettor does not        know the bet's final odds at the time the bet is made, as the        final odds are not known until the race starts and all the        betting has been completed. This fact creates at least three        problems for the bettor.        -   The bettor may end up making undesirable wagers. For            instance, the bettor may decide that a horse is a good bet            to win at odds of 4 to 1 or higher. The bettor may bet the            horse to win with a few minutes before the race starts when            the current indicative odds on the horse to win are 6 to 1.            However, just before the race starts, the final odds may go            to 2 to 1 (perhaps due to a large bet being made on the            horse just before the race starts) with the bettor being            unable to cancel his/her bet before the race starts. In this            example, the bettor has made a bet on a horse with odds that            the bettor views as undesirable.        -   The bettor may miss favorable betting opportunities if the            odds shift right before the start of the race. Say that a            horse has odds of 2 to 1 to win throughout the betting            period but immediately before the close of betting the odds            rise to 6 to 1 (perhaps due to large bets made on other            horses). There may not be enough time for the bettor to            observe the change and make a bet on the horse before the            race begins. Typically, indicative odds update with a            one-minute lag, so if a big bet is made less than one minute            before the race starts, then the indicative odds won't            change until after the racing association stops accepting            bets. In this case, it may be impossible for the bettor to            bet on the horse because betting will end before the bettor            can even observe the change in odds.    -   Since the indicative odds right before the start of the race are        most likely to reflect the final odds, many bettors monitor the        odds until the last possible moment and then hurry to make a bet        just before the race starts. This leads bettors to make crucial        betting decisions under time pressure with significant chance        for error.    -   Lack of Indicative Odds for Certain Bets. One common bet on        horse races is to bet on a horse to place. In this case, the        bettor wins if the selected horse finishes 1^(st) or 2^(nd). The        racing association does not provide indicative odds for place        bets during the betting period. Because of this, it is difficult        to determine whether or not to bet a horse to place. In fact,        the shrewd bettor will have to make somewhat complicated        calculations to approximate the expected odds and determine        whether a place bet is a good bet.    -   Lower Payouts Due to Separate and Small Pools. As mentioned        above, current parimutuel systems have different types of bets        segregated into different betting pools. For example, the win        and exacta pools are separate from each other even though these        pools could be combined. There are two negatives associated with        keeping these pools separate:        -   With separate pools, there is no aggregation of related bets            and so the bettor may find himself/herself moving the odds            against himself/herself when the bet is of a significant            size relative to the size of the pool.        -   With separate pools, bettors need to follow the changing            odds in multiple pools to search for good betting            opportunities. The time that the bettor spends doing this            takes away time from other activities, such as studying the            horses. Further, the bettor may miss good betting            opportunities because he/she is not able to monitor all the            possible bets from the multiple pools.    -   Inability to Make Certain Bets. In the current parimutuel        system, there are several types of interesting bets that cannot        be made directly including        -   Betting against a specific horse;        -   Betting on a horse to finish exactly 2^(nd);        -   Betting on a horse to finish exactly 2^(rd);        -   Betting on a horse to finish either 2^(nd) or 3^(rd), but            not 1^(st).            -   Approximating one of these bets requires making a large                number of bets where the chance of making an error in                submitting the bets correctly is high.    -   Stressful Betting Conditions. Currently, bettors may feel        obligated to make several bets at the last minute and monitor        several pools throughout the betting period. These conditions        reduce the enjoyment and increase the stress for bettors.

These imperfections of the current parimutuel system probably lessenbettors' enjoyment in betting on horse races. In addition, theseimperfections probably lead bettors to bet less often on horse races andbet less money when they do bet on horse races, which leads to lowerprofits for the racing association.

15.1.4 Gaming Against the House

Another widely used wagering technique is gaming against the house,which is used for many types of sports wagering.

For ease of explanation and illustrative purposes, let the wageringassociation be a casino and let the underlying event be the outcome of aspecific basketball game. A bettor might make a bet with the casino onwhich basketball team will win the game.

In gaming against the house, the casino's bookmaker(s) sets the odds andthen the bettor determines which team to bet on (if any) at these odds.The bettor submits to the casino the premium amount to be wagered. Thus,the bettor knows at the time the bet is made both the amount that he/shewill win if he/she wins the bet (based on the casino's odds) and theamount that he/she will lose (the premium amount) if he/she loses thebet.

If the team selected by the bettor wins the game, then the casino paysthe payout amount to the bettor and the bettor profits. If that teamloses the game, then the bettor loses the premium amount and the casinoprofits. This type of wagering is called gaming against the housebecause either the bettor profits or the casino (a.k.a. the “house”)profits. In this sense, the bettor is playing against the house.

A bookmaker tends to use two main principles for setting the odds ingaming against the house.

First, the bookmaker sets the odds in such a way that the casino expectsto make money over time. In other words, the bookmaker determines whatit thinks are true odds of a team winning a basketball game and thensets the odds for bettors to be lower than its estimate of the trueodds. By setting the odds at a lower number, the casino can expect tomake money over time by the law of averages. For a bet with true odds of1 to 1, the casino may set the odds at 10/11^(th)'S to 1. These arestandard odds for sports wagering, where a bettor typically puts up $11to win $10.

Second, the bookmaker sets odds such that the casino expects to makemoney regardless of the outcome or in the example above, which team winsthe basketball game. To achieve this, the bookmaker sets the odds suchthat some bettors bet on one team to win the basketball game and somebettors bet on the other team to win the basketball game. If thebookmaker splits the bettors successfully and sets the odds for each betat a lower number than its estimate (as discussed in the previousparagraph), then regardless of which team wins, the casino will end upreceiving more premium than the casino has to payout to winners, and sothe casino profits. In this case, the casino makes money regardless ofwhich outcome occurs and the casino has balanced its book. When its bookis not balanced, the casino may lose money if a certain team wins thegame.

If, during the betting period, the casino's book becomes unbalanced, thebookmaker may adjust the odds for all new bets on the basketball game inthe expectation and hope that new wagers will balance out the previouswagers. Any change in the odds made by the bookmaker does not impact theodds, premiums, and payouts of wagers that have already been made—thechange only impacts new wagers that are made. For instance, if earlybetting suggests that the casino will lose money if a specific teamwins, then the bookmaker may lower the odds for that team and increasethe odds for the other team for all new bets. By changing the odds inthis way, new bettors will be more likely than before to bet on the teamwith the increased odds and this will have the effect of more closelybalancing the casino's book. The fact that different bettors betting onthe same team to win may receive different odds, depending on the timethe bet is made, stands in contrast to parimutuel systems, where allbets on the same outcome receive the same odds, regardless of the timethe bet is made.

For more details on wagering on sports and a discussion on howbookmakers set odds, see section 5 of David Sklansky's Getting the Bestof It, Two Plus Two Publishing, 1993, Nevada and see the appendix inRichard Davies's and Richard Abram's Betting the Line, The Ohio StateUniversity Press, 2001, Ohio.

Gaming against the house is different than parimutuel wagering in twofundamental ways. First, in wagering against the house, the casino maymake or lose money depending on the outcome, whereas in parimutuelwagering, the racing association makes the same amount of moneyregardless of the outcome of the horse race. Second, in wagering againstthe house, the bettor knows the payout and the odds at the time the betis made. In contrast, in parimutuel wagering, the bettor does not knowthe final odds and the payout until well after the bet is made.

15.1.5 Short-Comings of Gaming Against the House

Gaming against the house can have disadvantages to the bettor such asthe following.

-   -   Poor Odds on Long Shots. The casino generally does not provide        close to fair odds on teams that are unlikely to win, presumably        because of the casino's oligopolistic position, concerns about        bettors having asymmetric information, and the casino's        risk-aversion. Thus, bettors have difficulty getting high odds        on teams unlikely to win. For instance, if a casino thinks a        team has odds of 20 to 1 of winning a basketball game, the        casino may only set the odds at 10 to 1. Because of this, a        bettor with information that a long-shot team has a good chance        of winning may be unable to capitalize on this wagering        opportunity. These concepts are discussed in more detail in        Alistair Bruce and Johnnie Johnson's paper, Investigating the        Roots of the Favourite-Longshot Bias: An Analysis of Decision        Making by Supply-and Demand-Side Agents In Parallel Betting        Markets, Journal of Behavior Decision Making 13, pages 413-430.    -   Betting Constraints. Bettors that want to make large bets may        not be able to bet their full amount as many casinos have a        maximum bet allowable at any one time.

In addition, gaming against the house can have some significantdisadvantages for the casino including the following.

-   -   Large Support Staff. The casino employs a large staff of        bookmakers to do research to set odds, monitor bets made, and        adjust odds over time to try and balance the casino's book.        Employing these people is a significant expense for many        casinos.    -   Losses from an Unbalanced Book. A casino can lose a large amount        of money if its book is unbalanced and if a specific team wins a        game. Large losses of this kind are an unappealing business risk        to a casino.

These disadvantages lower the attractiveness of sports wagering as abusiness for casinos.

15.1.6 Summary of the Invention

This invention is a method for wagering and gaming that should increasethe attractiveness of gaming to bettors and increase the profitabilityof casinos, horse and dog racing associations, and lottery organizationsthrough increased gaming participation. This invention has significantadvantages over current gaming systems such as parimutuel wagering andgaming against the house. The invention is referred to as the enhancedparimutuel system, since it builds on parimutuel methods.

Example embodiments of the invention involve the use of electronictechnologies, including computers, mathematical algorithms, computerprograms, and computerized databases for implementation. During thebetting period, the bets are entered into the computer as they are made.The invention allows for the calculation and display of indicative oddson all possible bets, just as is currently done at horse races. Afterthe betting period is over and all bets are

entered into the computer, an example embodiment computes the final oddson different bets and executes the maximum amount of premium. Final oddsare set and bets are executed so that regardless of the outcome of thehorse race, the amount in the parimutuel pool (net of fees) exactlyequals the amount to be paid-out to the holders of winning bets. Anexample embodiment uses the parimutuel framework (no risk to thewagering association) while allowing bettors to specify conditions underwhich their bets are filled.

In an example embodiment, all allowable bets are expressed as acombination of certain fundamental bets. Expressing bets in this way ispowerful: by expressing every bet as a combination of fundamental bets,every bet can be entered into the same parimutuel pool. The concept offundamental bets can be derived from the analytical approach called thestate space approach, which has been used in the financial academicliterature. For more detail, see Chi-fu Huang and Robert Litzenberger,Foundations for Financial Economics, 1988, Prentice Hall.

In an example embodiment, a bettor can specify the minimum or limit oddsthat the bettor is willing to accept for the bet to be executed. Forinstance, the bettor might bet $10 on a horse to win with limit odds of4 to 1 or higher. In this case, the bet is valid only if the final oddsfor the horse are 4 to 1 or higher. If the final odds are lower than 4to 1 (e.g., 3 to 1), then the bettor's bet will be cancelled and theracing association will return the premium to the bettor. Thus, bets areconditional bets in the sense that the bets will not be filled if theconditions specified are not met. It is worth noting that the conditionsonly relate to the final odds and do not in any way depend on theindicative odds during the betting period.

15.1.7 Invention Improvements

This invention provides the following benefits to bettors who makeparimutuel wagers, such as on horse races.

-   -   Limit Odds Bets Give Execution Control to the Bettor. The bettor        knows at the time a bet is made the lowest or “worst” possible        odds that he/she shall receive for the bet and the largest        premium that he/she shall pay.        -   The bettor will not make undesirable wagers if the odds on a            horse of interest drop late in the betting period. If the            odds on the horse become unfavorable, then the bet will not            be executed and the racing association returns the premium            to the bettor.        -   The bettor will not miss favorable betting opportunities if            the odds shift favorably immediately before the start of the            race. The bettor can make bets on a horse of interest and            they will be executed automatically if the final odds are            favorable.        -   The bettor does not need to monitor odds throughout the            betting period searching constantly for attractive betting            opportunities. The bettor can enter the bets that are            attractive to the bettor and the invention will            automatically execute the wagers if the conditions are met.            There is no need for last minute split second decisions            using the invention.    -   Indicative Odds for All Bets. The invention provides indicative        odds and payouts to the bettors for all bets.    -   One Combined Large Pool. Using the methodology based on        fundamental bets, all bets are combined into one pool, which        offers several benefits for the bettor.        -   The bettor will enjoy greater liquidity, as his/her own bets            will impact the final odds less than they would in the            current parimutuel system.        -   With one combined pool, bettors no longer need to monitor            multiple pools to determine the pool to enter a bet.    -   Ability to Make New Types of Bets With Liquidity. The invention        allows the racing association to offer new bets to the bettor        including        -   Betting against a specific horse;        -   Betting on a horse to finish exactly 2^(nd);        -   Betting on a horse to finish exactly 3^(rd);        -   Betting on a horse to finish either 2^(nd) or 3^(rd), but            not 1^(st).    -   These bets can be made in one step without the need to make a        large number of bets. Further, because these bets will be        entered into the combined pool, relatively large bets may not        significantly affect the odds for these new bets.    -   Greater Efficiency and Increased Enjoyment. Shrewd bettors may        experience greater efficiency in betting on horses since bets        will no longer need to be made just before the race starts.        Further, the bettor will not have to monitor many different        pools throughout the betting period. The bettor may make his/her        bets with their limit odds at any time during the betting        period. The invention frees the bettor up to do research or        other activities during the betting period.    -   No Added Complexity for Bettors. The invention can be        implemented in a way that will be nearly transparent to bettors        on horse races and so a bettor can submit a bet in almost an        identical fashion to current methods. Thus, the invention        requires limited or no change in procedures for current bettors        on horse races.

In addition, example embodiments provide bettors with the followingbenefits compared to wagering against the house.

-   -   Higher Odds for Long Shots. Bettors will likely be able to        receive higher odds on long shots, as odds will be determined        purely by the amount in the betting pools, not by limits set by        the casino.    -   No Maximum Bet Size. The invention eliminates the casino's        business need to have a maximum bet size so bettors will be able        to make sports bets for any amount desired.    -   Equal Footing. In gaming against the house, bettors may believe        that they are at a disadvantage, as they are up against a        sophisticated, well-informed, and deep-pocketed opponent, namely        the casino. In enhanced parimutuel wagering, bettors are        effectively betting against other similar participants. Because        of this, bettors may find wagering in a parimutuel setting        preferable to wagering against the house.

Example embodiments provide benefits to wagering organizations such asracing associations and casinos because the improved features willlikely lead to more betting and increased profit for theseorganizations. This invention provides the following additional benefitsto gaming organizations.

-   -   No Employees Required for Odds-Making. The casino will not need        to employ staff when using an example embodiment to determine        odds, monitor the casino's book or vary the odds due to betting        imbalances, as the invention performs these functions        automatically.    -   The Casino's Book is Always Balanced. The casino will not have        any losses or risks from an unbalanced book. The invention keeps        the casino's book balanced and the risk equal to zero.    -   The Casino's Profit is Clear and Easy to Compute. The casino can        set the percentage of premium bet or the percentage of total        payouts to take as profit for itself, varying the percentage        based on the type of bet or the underlying event. Once this        percentage is known, this profit will depend directly on the        total amount bet: the more premium that is bet, the higher the        casino's profit.

Section 15.2 builds on the description of enhanced parimutuel wageringand adds the mathematical underpinnings to this approach.

15.2 Details and Mathematics of Enhanced Parimutuel Wagering

This section describes the mathematical details of example embodimentsusing a horse-racing example for illustrative purposes.

15.2.1 Set-Up

In an example embodiment, the wagering association first determines afuture event for wagering. This underlying event will have multipleoutcomes that are measurable. For example, the wagering association maybe a racing association that runs a three horse race with the horsesnumbered 1, 2, and 3.

The wagering association determines the types of wagers that bettorswill be allowed to make. For the horse race, assume that the wageringassociation allows wagers based on the horse that finishes 1^(st) andthe horse that finishes 2^(nd). Assume that all bets are in U.S. dollarsand that all three horses finish the race.

The wagering association sets a time period or betting period duringwhich bets and premium amounts will be received. For a horse race, thebetting period will often begin early on the day of the race and endwith the race's start. (In certain cases, the wagering association maywish to set up more than one betting period per underlying event. Forinstance, for wagering on a widely followed horse race, the wageringassociation may wish to have separate betting periods each day on theseveral days leading up to and including race day. For each separatebetting period, there will be a separate parimutuel pool and differentfinal odds resulting.) Typically, the wagering association allowsbettors to collect their payouts on the date that the underlying eventoccurs.

At a horse race, payouts are typically available within a few minutes ofthe end of the race, after the results of the race are official.

15.2.2 The Fundamental Outcomes and the Fundamental Bets

After determining the different types of wagers that bettors will beallowed to make, the wagering association determines the fundamentaloutcomes, which must satisfy two properties:

-   -   (1) The fundamental outcomes represent a mutually exclusive and        collectively exhaustive set of the outcomes from the underlying        event;    -   (2) All winning outcomes of wagers are combinations of these        fundamental outcomes.

These fundamental outcomes are derived from states in the financeliterature. The term “fundamental” is borrowed from finance: in finance,state claims are often referred to as fundamental contingent claims.

Let S denote the number of fundamental outcomes associated with thetypes of wagers allowed by the wagering association. Let s index thefundamental outcomes, so s=1, 2, . . . , S. For the three horse race,the number of fundamental outcomes S equals six and these outcomes arelisted in Table 15.2.2A.

TABLE 15.2.2A The fundamental outcomes for a three horse race withwagering on horses to finish in the first two places. FundamentalOutcome 1^(st) Place Finisher 2^(nd) Place Finisher 1 Horse 1 Horse 2 2Horse 1 Horse 3 3 Horse 2 Horse 1 4 Horse 2 Horse 3 5 Horse 3 Horse 1 6Horse 3 Horse 2

It is worth emphasizing that the fundamental outcomes for an eventdepend on the type of wagers that the wagering association allows. In athree horse race with wagering on the 1^(st) and 2^(nd) place finishers,there are six fundamental outcomes. However, if the wagering associationallows wagers only on the winner of the three horse race, then there areonly three fundamental outcomes: horse 1 finishes first, horse 2finishes first, and horse 3 finishes first.

Each fundamental outcome is associated with a fundamental bet, where thesth fundamental bet pays out $1 if and only if the sth fundamentaloutcome occurs. Exactly one fundamental bet will payout based on theunderlying event, since the fundamental outcomes are a mutuallyexclusive and collectively exhaustive set of outcomes. The number offundamental bets is equal to S, the number of fundamental outcomes, andthe fundamental bets will again be indexed by s with s=1, 2, . . . , S.Just as every outcome can be represented as a combination of thefundamental outcomes, every bet can be broken into a combination offundamental bets. Because of this, every bet can be entered into thesame parimutuel pool, a powerful approach for aggregating liquidity.

Table 15.2.2B displays the six fundamental bets for a three horse racewith wagers on horses to finish 1^(st) and 2^(nd).

TABLE 15.2.2B Fundamental bets for a three horse race with wagering onhorses to finish in the first two places. Outcome/Fundamental Bet sSpecified Outcome for Fundamental Bet 1 Horse 1 finishes 1^(st), horse 2finishes 2^(nd) 2 Horse 1 finishes 1^(st), horse 3 finishes 2^(nd) 3Horse 2 finishes 1^(st), horse 1 finishes 2^(nd) 4 Horse 2 finishes1^(st), horse 3 finishes 2^(nd) 5 Horse 3 finishes 1^(st), horse 1finishes 2^(nd) 6 Horse 3 finishes 1^(st), horse 2 finishes 2^(nd)

One can express, for example, bets on a horse to win as combinations ofthese fundamental bets.

For example, a bet on horse 1 to win can be expressed as “horse 1 winsand any other horse finishes 2^(nd).” If horse 1 wins, then either horse2 or horse 3 must finish 2^(nd). Thus horse 1 wins the race if and onlyif either of the following outcomes occurs:

-   -   1) Horse 1 wins and horse 2 finishes 2^(nd); or    -   2) Horse 1 wins and horse 3 finishes 2^(nd).        Thus a bet on horse 1 to win is a combination of fundamental        bets 1 and 2. Similarly, one can express bets on horse 2 or 3 to        win, a horse to place, a horse to finish 2^(nd), exacta bets,        and quinella bets as combinations of these six fundamental bets.

15.2.3 Opening Bets

Before the wagering association accepts bets during the betting period,the wagering association may enter bets for each of the S fundamentaloutcomes referred to as the opening bets. Let the sth opening bet payoutif and only if the sth fundamental outcome occurs, and let θ_(s) be theamount of that opening bet for s=1, 2, . . . , S. An example embodimentmay require

θ_(s)>0 s=1, 2, . . . , S  15.2.3A

Opening bets ensure that the final prices and odds are unique. See, forexample, the unique price proof in section 7.11.

The wagering association may wish to keep the amount of opening betssmall to limit the wagering association's risk in the race. Forinstance, for the three horse race, the wagering association might enter$1 in premium for each outcome, i.e. θ_(s)=1 for s=1, 2, . . . , 6.Alternatively, the wagering association may wish to follow theobjectives discussed in section 11.4.1 for determining the opening bets.

15.2.4 Bets from Customers

During the betting period, the wagering association accepts bets frombettors. In making a bet, first, the bettor specifies the type of betand the horse(s). The bettor may specify the maximum premium amount thatthe bettor wishes to spend. This is called a premium bet. Alternatively,the bettor may specify the maximum payout that the bettor receives ifthe bet wins, and this bet is referred to as a payout bet.

Next, the bettor specifies the minimum or limit odds that the bettor iswilling to accept for the bet to be executed. For instance, the bettormight bet $10 on a horse to win with limit odds of 4 to 1 or higher. Inthis case, the bet is valid only if the final odds for the horse are 4to 1 or higher. If the final odds are lower than 4 to 1, then thebettor's bet will be cancelled and the wagering association will returnthe premium (if submitted) to the bettor. Currently in betting on horseraces, bettors do not specify the limit odds. In an example embodiment,this case can be handled by setting the limit odds to 0 (in thefinancial markets, this would be called an order at the market) and inthis case the bet is executed regardless of the odds.

For notation, let J be the number of bets made by bettors in the bettingperiod. Let o_(j) denote the limit odds per $1 of premium bet for j=1,2, . . . , J. In the example described in the previous paragraph,o_(j)=4. Let u_(j) denote the premium amount requested if bet j is apremium bet, and let r_(j) denote the maximum payout amount requested ifbet j is a payout bet.

15.2.5 Representing Bets Using the Fundamental Bets

In an example embodiment, the winning outcomes from a bet can be relatedto specific fundamental outcomes. For j=1, 2, . . . , J, let a_(j) be a1 by S row vector where the sth element of a_(j) is denoted by a_(j,s).Here, a_(j,s) is proportional to bet j's requested payout if fundamentaloutcome s occurs. If a_(j,s) is 0, then the bettor requests no payout iffundamental outcome s occurs. If a_(j,s) is greater than 0, then thebettor requests a payout if fundamental outcome s occurs. Forsimplicity, restrict a_(j) as follows

min{a_(j,1), a_(j,2), . . . , a_(j,s)}=0 for j=1, 2, . . . , J  15.2.5A

max {a_(j,1), a_(js)}=1 for j=1, 2, . . . , J 15.2.5BCondition 15.2.5A requires that the bettor has a least one outcome inwhich the bet will receive no payout. Condition 15.2.5B is a scalingcondition that requires the maximum value payout per unit of bet to beequal to 1. The vector a_(j) will be referred to as the weighting vectorfor bet j.

One can construct a_(j) for different types of bets. Recall from section15.2.2 that a bet on horse 1 to win is a combination of fundamental bets1 and 2 and thus in this case

a_(j)=[1 1 0 0 0 0]  15.2.5C

Similarly, a bet on horse 2 to win is a combination of fundamental bets3 and 4, and thus

a_(j)=[0 0 1 1 0 0]  15.2.5D

A bet on horse 3 to win is a combination of fundamental bets 5 and 6,and therefore

a_(j)=[0 0 0 0 1 1]  15.2.5E

In an example embodiment, the wagering association can accept andprocess bets against specific outcomes or sell bets in enhancedparimutuel wagering. For example, consider a bettor who wants to makeprofit if the 1 horse does not win the race and that bettor is willingto lose premium if the 1 horse wins the race. This bet is equivalent tobetting against the 1 horse or in financial parlance “selling short the1 horse.” This is a combination of fundamental bets 3, 4, 5, and 6 andthus

a_(j)=[0 0 1 1 1 1]  15.2.5F

A bet on horse 2 to win and horse 3 to finish second is equivalent tofundamental bet 4, and so in this case

a_(j)=[0 0 0 1 0 0]  15.2.5G

A bet on horse 3 to win and horse 2 to finish second is equivalent tofundamental bet 6, and so

a_(j)=[0 0 0 0 0 1]  15.2.5H

Similarly, a bet on horse 3 to place (finish 1^(st) or 2^(nd)) is acombination of fundamental bets 2, 4, 5, and 6 and thus

a_(j)=[0 1 0 1 1 1]  15.2.51

A bettor may desire different payouts depending on which outcome occurs.For example, the bettor may wish to make twice the payout if horse 2wins versus if horse 2 finishes 2^(nd). In this case,

a_(j)=[0.5 0 1 1 0 0.5]  15.2.5J

The vector a_(j) can be determined for other bets as well.

15.2.6 Pricing Bets Using the Prices of the Fundamental Bets

Let p_(s) denote the final price of the sth fundamental bet with apayout of $1. Based on the price for a $1 payout, the odds for thatfundamental bet are (1/p_(s))−1 to 1.

Mathematically, the wagering association may require that

$\begin{matrix}{{{p_{s} > {0\mspace{14mu} s}} = 1},2,\ldots \mspace{14mu},S} & {15.2{.6}A} \\{{\sum\limits_{s = 1}^{S}p_{s}} = 1} & {15.2{.6}B}\end{matrix}$

Here, the wagering association requires that the prices of thefundamental bets are positive and sum to one.

The wagering association may determine the price of each bet using theprices of the fundamental bets as follows. Let π_(j) denote the finalprice for a $1 payout for bet j. For simplicity of exposition, assumehere that the wagering association does not charge fees (see section 7.8for a discussion of fees). Then, the price for bet j is

$\begin{matrix}{\pi_{j} \equiv {\sum\limits_{s = 1}^{S}{a_{j,s}p_{s}}}} & {15.2{.6}C}\end{matrix}$

The price of each bet is the weighted sum of the prices of thefundamental bets. The final odds to $1 for bet j are given by

ω_(j)=(1/π_(j))−1  15.2.6D

As in simple parimutuel systems, all customers with the same bet receivethe same odds if they are filled on the bet, regardless of their limitodds.

15.2.7 Determining Fills Using Limit Odds

In an example embodiment, the bets can be filled by comparing the limitodds and the final odds. For notation, let x_(j) be equal to the filledpayout amount and let v_(j), denote the filled premium for bet j.

The logic for a premium bet is as follows. If the final odds ω_(j) areless than the limit odds o_(j), then the filled premium v_(j) equals 0as the bet is not executed. If the final odds ω_(j) are equal to thelimit odds o_(j), then 0≦v_(j)≦u_(j). In this case, the bet may bepartially executed. If the final odds ω_(j) are higher than the limitodds o_(j), then v_(j) equals u_(j) and the bet is fully executed. Tosummarize this logic for a premium bet

ω_(j)<o_(j)→v_(j)=0

ω_(j)=o_(j)→0≦v_(j)≦u_(j)

ω_(j)>o_(j)→v_(j)=u_(j)  15.2.7A

Once the filled premium v_(j) is determined for the premium bet, thefilled payout x_(j) for this bet can be computed by the formula

$\begin{matrix}{x_{j} = \frac{v_{j}}{\pi_{j}}} & {15.2{.7}B}\end{matrix}$

For a payout bet, if the final odds ω_(j) are less than the requestedodds o_(j), then x_(j) equals 0. If a payout bet has requested oddso_(j) equal to the final odds ω_(j), then 0≦x_(j)≦r_(j). If, for bet j,the final odds ω_(j) are higher than the requested odds o_(j), thenx_(j) equals r_(j). To summarize this logic for a payout bet

ω_(j)<o_(j)→x_(j)=0

ω_(j)=o_(j)→0≦x_(j)≦r_(j)

ω_(j)>o_(j)→x_(j)=r_(j)  15.2.7A

Once the filled payout amount is determined, then the filled premiumv_(j) is determined by the formula

v_(j)=x_(j)π_(j)  15.2.7D

The logic in equations 15.2.7A and 15.2.7C is similar to the logicdescribed in equations in sections 7.7, 7.11, and 11.4.4. In thoseequations, however, the limit price w_(j) is compared to the final priceπ_(j). Since the limit price and the limit odds are related via

w _(j)=(1/o _(j))−1  15.2.7E

and the final price π_(j) and the final odds are related via equation15.2.6D, these equations can be derived from the earlier equations.

In an example embodiment, ω_(j), the final odds per $1 of bet j, are notnecessarily equal to the bettor's limit odds o_(j). In an exampleembodiment, every bet of a particular type with limit odds less than thefinal odds receives the same final odds.

15.2.8 Final Pricing Conditions and Self-Hedging

Let M denote the total premium paid in the betting period, which can becomputed as follows

$\begin{matrix}{M \equiv {\left( {\sum\limits_{j = 1}^{J}{x_{j}\pi_{j}}} \right) + {\sum\limits_{s = 1}^{S}\theta_{s}}}} & {15.2{.8}A}\end{matrix}$

or equivalently as

$\begin{matrix}{M \equiv {\left( {\sum\limits_{j = 1}^{J}v_{j}} \right) + {\sum\limits_{s = 1}^{S}\theta_{s}}}} & {15.2{.8}B}\end{matrix}$

based on formula 15.2.7D.Next, note that a_(j,s)x_(j) is the amount of fundamental bet s used tocreate bet j. Define y_(s) as

$\begin{matrix}{y_{s} \equiv {\sum\limits_{j = 1}^{J}{a_{j,s}x_{j}}}} & {15.2{.8}C}\end{matrix}$

for s=1, 2, . . . , S. Here, y_(s) is the aggregate filled amount acrossall bets that payout if fundamental outcome s occurs. Note that sincethe a_(j,s)'s are non-negative (equation 15.2.5A), and the x_(j)'s arenon-negative, y_(s) will also be non-negative.

The self hedging condition is the condition that the total premiumcollected is exactly sufficient to fund the payouts to winning bettors.The self-hedging condition can be written as

$\begin{matrix}{{{y_{s} + \frac{\theta_{s}}{p_{s}}} = {{M\mspace{14mu} s} = 1}},2,\ldots \mspace{14mu},S} & {15.2{.8}D}\end{matrix}$

The wagering association takes on risk to the underlying only throughP&L in the opening bets.

Equation 15.2.8D relates y_(s), the aggregated filled amount of the sthfundamental bet, and p_(s), the price of the sth fundamental bet. For Mand θ_(s) fixed, the greater y_(s), then the higher p_(s) and the higherthe prices (or equivalently, the lower the odds) of bets that pay out ifthe sth fundamental bet wins. Similarly, the lower the bet payoutsy_(s), then the lower p_(s) (or equivalently, the higher the odds) andthe lower the prices of bets that pay out if the sth fundamental betwins. Thus, in this pricing framework, the demand for a particularfundamental bet is closely related to the price for that fundamentalbet.

15.2.9 Maximizing Premium to Determine the Final Fills and the FinalOdds

In determining the final fills and the final odds, the wageringassociation may seek to maximize the total filled premium M subject tothe constraints described above. Combining all of the above equations toexpress this mathematically gives the following set of equations toengage in a demand-based valuation of each of the fundamental bets, andhence determine final odds, filled premiums and payouts for wagers inthe betting pool

$\begin{matrix}\begin{matrix}{{maximize}\mspace{14mu} M} & \; & \; \\{{subject}\mspace{14mu} {to}} & \; & \; \\{{\left. 1 \right)\mspace{14mu} 0} < p_{s}} & {{s = 1},2,\ldots \mspace{14mu},S} & \; \\{{\left. 2 \right)\mspace{14mu} {\sum\limits_{s = 1}^{S}p_{s}}} = 1} & \; & \; \\{{\left. 3 \right)\mspace{14mu} \pi_{j}} \equiv {\sum\limits_{s = 1}^{S}{a_{j,s}p_{s}}}} & {{j = 1},2,\ldots \mspace{14mu},J} & \; \\{{\left. 4 \right)\mspace{14mu} {If}\mspace{14mu} {bet}\mspace{14mu} j\mspace{14mu} {is}\mspace{14mu} a\mspace{14mu} {premium}\mspace{14mu} {bet}},{then}} & \; & \; \\{{{{\left. \mspace{45mu} {4A} \right)\mspace{14mu} \omega_{j}} < o_{j}}->v_{j}} = 0} & \; & \; \\{{\left. \mspace{45mu} {4\; B} \right)\mspace{14mu} \omega_{j}} = {o_{j}->{0 \leq v_{j} \leq u_{j}}}} & \; & \; \\{{{{\left. \mspace{45mu} {4C} \right)\mspace{14mu} w_{j}} > o_{j}}->v_{j}} = u_{j}} & \; & \; \\{{\left. 5 \right)\mspace{14mu} {If}\mspace{14mu} {bet}\mspace{14mu} j\mspace{14mu} {is}\mspace{14mu} a\mspace{14mu} {payout}\mspace{14mu} {bet}},{then}} & \; & \; \\{{{{\left. \mspace{45mu} {5A} \right)\mspace{14mu} \omega_{j}} < o_{j}}->x_{j}} = 0} & \; & \; \\{{\left. \mspace{45mu} {5B} \right)\mspace{14mu} \omega_{j}} = {o_{j}->{0 \leq x_{j} \leq r}}} & \; & \; \\{{{{\left. \mspace{45mu} {5C} \right)\mspace{14mu} \omega_{j}} > o_{j}}->x_{j}} = r_{j}} & \; & \; \\{{\left. 6 \right)\mspace{14mu} M} \equiv {\left( {\sum\limits_{j = 1}^{J}v_{j}} \right) + {\sum\limits_{s = 1}^{S}\theta_{s}}}} & \; & \; \\{{\left. 7 \right)\mspace{14mu} y_{s}} \equiv {\sum\limits_{j = 1}^{J}{a_{j,s}x_{j}}}} & {{s = 1},2,\ldots \mspace{14mu},S} & \; \\{{{\left. 8 \right)\mspace{14mu} y_{s}} + \frac{\theta_{s}}{p_{s}}} = M} & {{s = 1},2,\ldots \mspace{14mu},S} & \;\end{matrix} & {15.2{.9}A}\end{matrix}$

This maximization of M can be solved using the mathematical programmingmethods of section 7.9. Based on this maximization, the wageringassociation determines the final fills and the final odds. During thebetting period, the wagering association can display indicative odds andindicative fills calculated based on the assumption that no more betsare received during the betting period.

15.3 Horse-Racing Example

As an illustrative numerical example, consider, as before, a three horserace with the horses numbered 1, 2, and 3. For this horse race, thewagering association allows wagers based on the horse that finishes1^(St) and the horse that finishes 2^(nd) and all bets are in U.S.dollars. There are six fundamental bets, so S=6. The fundamental betsare as listed in column two of Table 15.3A (also listed previously inTable 15.2.2A). As shown in column three, the wagering associationenters $1 in premium for each of the fundamental bets so θ_(s)=1 fors=1, 2, . . . , 6.

TABLE 15.3A Fundamental bets for a three horse race with wagering on thefirst two finishers. Outcome/ Fundamental Fundamental Specified Outcomefor Fundamental Bet Amount Bet s Bet θ_(s) 1 Horse 1 finishes 1^(st),horse 2 finishes 2^(nd) $1 2 Horse 1 finishes 1^(st), horse 3 finishes2^(nd) $1 3 Horse 2 finishes 1^(st), horse 1 finishes 2^(nd) $1 4 Horse2 finishes 1^(st), horse 3 finishes 2^(nd) $1 5 Horse 3 finishes 1^(st),horse 1 finishes 2^(nd) $1 6 Horse 3 finishes 1^(st), horse 2 finishes2^(nd) $1

During the betting period, six bets are submitted by customers so J=6.Table 15.3B shows the details of these bets. The first, second, andthird bets are for horses 1, 2, and 3 to finish first, respectively. Thefourth bet is a bet that horse one does not finish first. Bets five andsix are exacta bets. (Note that these bets are the first six betsdiscussed in section 15.2.5.) The bet descriptions are in column two ofTable 15.3B. Column three of this table shows the limit odds (submittedby bettors) for these bets. All of these bets are premium bets (asopposed to payout bets) and column four shows the premium amountrequested for each of these bets. The remaining columns of Table 15.3Bshow the weights for these six bets.

TABLE 15.3B The bets and weights for the three horse race with wageringon the first two finishers. Limit Premium Odds to 1 Amount Bet j BetDescription o_(j) Requested u_(j) a_(j,1) a_(j,2) a_(j,3) a_(j,4)a_(j,5) a_(j,6) 1 Horse 1 finishes first 4 to 1 5 1 1 0 0 0 0 2 Horse 2finishes first 1 to 1 100 0 0 1 1 0 0 3 Horse 3 finishes first 1.5 to1   40 0 0 0 0 1 1 4 Horse 1 doesn't finish first 1 to 1 50 0 0 1 1 1 15 Horse 2 finishes first, 9 to 1 10 0 0 0 1 0 0 horse 3 finishes second6 Horse 3 finishes first, 3 to 1 25 0 0 0 0 0 1 horse 2 finishes second

Based on these bets, one can solve equation 15.2.9A for the final odds,filled premium amounts, and the prices of the fundamental bets. Table15.3C and Table 15.D show these results.

TABLE 15.3C Final prices of fundamental bets for a three horse race withwagering on the first two finishers. Total Outcome/ Bet AggregateOpening Bet Outcome Fundamental Price Final Customer Payout Payout Bet sp_(s) Odds to 1 Payouts y_(s) θ_(s)/p_(s) y_(s) + θ_(s)/p_(s) 1 0.05 19to 1  $50 $20 $70 2 0.05 19 to 1  $50 $20 $70 3 0.25 3 to 1 $66 $4 $70 40.25 3 to 1 $66 $4 $70 5 0.15 5.67 to 1   $63.33 $6.67 $70 6 0.25 3 to 1$66 $4 $70

TABLE 15.3D The final odds and fills for the three horse race withwagering on the first two finishers. Customer Customer Final FinalFilled Filled Odds Price Bet j Bet Description Premium v_(j) Payoutx_(j) ω_(j) to 1 of Bet π_(j) 1 Horse 1 finishes first 5 50 9 to 1 0.1 2Horse 2 finishes first 33 66 1 to 1 0.5 3 Horse 3 finishes first 25.3363.33 1.5 to 1   0.4 4 Horse 1 doesn't finish first 0 0 0.11 to 1   0.95 Horse 2 finishes first, horse 3 0 0 3 to 1 0.25 finishes second 6Horse 3 finishes first, horse 2 0.67 2.67 3 to 1 0.25 finishes secondIt is instructive to verify that the numerical values in these tablesmatch the eight equilibrium conditions set forth in equation 15.2.9A.

Column two of Table 15.3C shows the prices of the fundamental bets. Itis not hard to check that the prices of the fundamental bets arepositive and sum to one, satisfying conditions one and two,respectively.

To verify condition three, note that for j=1, condition three can bewritten as

π₁ =a _(1,1) p ₁ +a _(1,2) p ₂ +a _(1,3) p ₃ +a _(1,4) p ₄ +a _(1,5) p ₅+a _(1,6) p ₆  15.3A

Observing row one of Table 15.3B, note that

a_(1,3)=a_(1,4)=a_(1,5)=a_(1,6)=0  15.3B

Therefore,

π₁ =a _(1,1) p ₁ +a _(1,2) p ₂ +a _(1,3) p ₃ +a _(1,4) p ₄ +a _(1,5) p ₅+a _(1,6) p ₆ =a _(1,1) p ₁ +a _(1,2) p ₂  15.3C

Note that

a _(1,1) p ₁ +a _(1,2) p ₂=1×(0.05)+1×(0.05)=0.1  15.3D

and thus π₁=0.1 satisfies condition three. Condition three can also beverified for j=2, 3, . . . , 6.

Next, one can check that conditions 4A, 4B, and 4C are satisfied for thesix premium bets. For example, for bet j=1, note that the market oddsw₁=9 are higher than the limit odds o₁=4, and thus v₁=u₁=5. Thus thepremium fill for bet 1 satisfies condition 4C. For j=2, note that themarket odds equal the limit odds, i.e. 6)₂=o₂=1. For this bet, thefilled premium v₂=33 is between 0 and the requested premium u₂, whichequals 100. Thus the premium fill for bet 2 satisfies condition 4B. Onecan check that this logic is satisfied for bets three through six. Notethat conditions 5A, 5B, and 5C (the conditions for payout bets) do notneed to be verified since all bets in this example are premium bets.

Once the filled premium amounts are known, the payout amounts can becomputed with equation 15.2.7B. For example, for the first bet,v₁/π₁=(5/0.1)=50, which equals x₁, the payout amount. This can beverified for the other bets as well, confirming that column four ofTable 15.3D satisfies equation 15.2.7B.

Condition six computes the total premium paid in the betting period. Inthis case J=6 and S=6 so

$\begin{matrix}{M \equiv {\left( {\sum\limits_{j = 1}^{6}v_{j}} \right) + {\sum\limits_{s = 1}^{6}\theta_{s}}}} & {15.3E}\end{matrix}$

Here, the total premium is the sum of the premium paid by customers andthe sum of the opening bets. Using column three of Table 15.3D, notethat

v ₁ +v ₂ +v ₃ +v ₄ +v ₅ +v ₆=5+33+25.33+0+0+0.67=64  15.3F

Further, note that the total amount of the opening bets equals 6.Therefore, M=70, where 70 equals 64 plus 6.

Next, to verify the aggregate customer payouts y_(s) for conditionseven, consider s=1. In this case, condition seven simplifies to

y ₁ =a _(1,1) x ₁ +a _(2,1) x ₂ +a _(3,1) x ₃ +a _(4,1) x ₄ +a _(5,1) x₅ +a _(6,1) x ₆  15.3G

Now,

a _(1,1) x ₁ +a _(2,1) x ₂ +a _(3,1) x ₃ +a _(4,1) x ₄ +a _(5,1) x ₅ +a_(6,1) x ₆ =a _(1,1) x ₁  15.3H

since

a_(2,1)=a_(3,1)=a_(4,1)=a_(5,1)=a_(6,1)=0  15.3I

by column five of Table 15.3B. Therefore,

y ₁ =a _(1,1) x ₁=1×(50)=50  15.3J

Similarly, y₂, y₃, . . . , y₆ can be computed, verifying condition sevenand the values in column four of Table 15.3C.

To verify condition eight, the self-hedging condition, set s=1. In thiscase, condition eight can be written as

$\begin{matrix}{{y_{1} + \frac{\theta_{1}}{p_{1}}} = M} & {15.3K}\end{matrix}$

As shown in row one of Table 15.3C,

$\begin{matrix}{{50 + \frac{1}{0.05}} = 70} & {15.3L}\end{matrix}$

Condition eight can also be verified for s=2, 3, . . . , 6.

Thus, the eight conditions of equation 15.2.9A are satisfied in theexample considered here.

15.4 Additional Examples of Enhanced Parimutuel Wagering

This section provides several applications of enhanced parimutuelwagering. Section 15.4.1 discusses a horse-racing example. Section15.4.2 applies enhanced parimutuel wagering to gaming typically doneagainst the house. Section 15.4.3 examines a lottery example.

15.4.1 Using Enhanced Parimutuel Wagering for Horse-Racing

Consider a four horse race where the racing association offers bets onthe first three horses to finish. There are 24 fundamental outcomes forthis race and so S equals 24. Table 15.4.1A lists these fundamentaloutcomes.

TABLE 15.4.1A The 24 fundamental outcomes for a four horse race withwagering on the first three finishers. Fundamental 1^(st) Place 2^(nd)Place 3^(rd) Place Outcome Finisher Finisher Finisher 1 1 2 3 2 1 2 4 31 3 2 4 1 3 4 5 1 4 2 6 1 4 3 7 2 1 3 8 2 1 4 9 2 3 1 10 2 3 4 11 2 4 112 2 4 3 13 3 1 2 14 3 1 4 15 3 2 1 16 3 2 4 17 3 4 1 18 3 4 2 19 4 1 220 4 1 3 21 4 2 1 22 4 2 3 23 4 3 1 24 4 3 2More generally, in a horse race with h horses where the wageringassociation offers bets on the 1^(st) d horses to finish, the number offundamental outcomes is

$\begin{matrix}{S = {\frac{h!}{\left( {h - d} \right)!} = {h \times \left( {h - 1} \right) \times \ldots \times \left( {h - d + 1} \right)}}} & {15.4{.1}A}\end{matrix}$

where “!” denotes the factorial function. In the example here, h equals4 and d equals 3 and so S=4×3×2=24 and equation 15.4.1A holds. If thewagering association offers bets on all but one horse to finish, thend=h−1 and the number of fundamental outcomes is S=h!. If the wageringassociation offers bets on all the horses to finish, then d=h and thenumber of fundamental outcomes is also S=h!.

The next discussion shows how to map bets into the specified outcomesabove using the a_(j) vector introduced in section 15.2.5.

Betting on a Horse to Win. Using Table 15.4.1A, observe that horse 2wins if horse 2 finishes 1^(st) and any of the remaining horses finish2^(nd) and 3^(rd), which corresponds to outcomes 7 through 12. Thus thea_(j) vector equals 1 for fundamental outcomes 7 through 12 and 0otherwise.

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0

Betting on a Horse to Finish 2^(nd). To bet on horse 3 to finish 2^(nd)requires specifying outcomes where horse 3 finishes 2^(nd) and any ofthe remaining horses finish 1^(st) and 3^(rd). These events correspondto fundamental outcomes 3, 4, 9, 10, 23, and 24. Thus, a_(j) is asfollows

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1

Betting on a Horse to Place. To bet on horse 3 to place is to bet thathorse 3 will finish 1^(st) or 2^(nd). For this bet to win requiresoutcomes where

-   -   (1) Horse 3 finishes 1^(st) and any of the remaining horses        finish 2^(nd) and 3^(rd);    -   (2) Horse 3^(rd) finishes 2^(nd) and any of the remaining horses        finish 1^(st) and 3^(rd);        The 1^(st) condition is met by fundamental outcomes 13 through        18 and the 2^(nd) condition is met by fundamental outcomes 3, 4,        9, 10, 23, 24. Thus, a_(j) is as follows

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1

Betting on a Horse to Finish 3^(rd). To bet on horse 1 to finish 3^(td)requires specifying outcomes where horse 1 finishes 3^(rd) and anyremaining horses finish 1^(st) and 2^(nd), corresponding to fundamentaloutcomes 9, 11, 15, 17, 21, and 23. Thus, a_(j) is as follows

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 0

Betting on a Horse to Show. To bet on horse 1 to show (finish 1^(st),2^(nd) or 3^(rd)) requires the following outcomes

-   -   (1) Horse 0.1 finishes 1^(st) and any remaining horses finishes        2^(nd) and 3^(rd);    -   (2) Horse 1 finishes 2^(nd) and any remaining horses finishing        1^(st) and 3^(rd);    -   (3) Horse 1 finishes 3^(rd) and any remaining horses finishing        1^(st) and 2^(nd).        The 1^(st) condition is met by fundamental outcomes 1 through 6,        the 2″ condition is met by fundamental outcomes 7, 8, 13, 14,        19, 20 and the 3^(rd) condition is met by fundamental outcomes        9, 11, 15, 17, 21, and 23. Thus, for this bet, a_(j) is as        follows

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0

Betting on an Exacta Combination. To win an exacta bet requiresselecting the horse that finishes 1^(st) and the horse that finishes2^(nd) in the correct order. To bet the ¾ exacta is equivalent toselecting the following outcomes: horse 3 wins, horse 4 finishes 2^(nd),and any of the remaining horses finish 3^(rd). This corresponds tofundamental outcomes 17 and 18. Thus, a_(j) is as follows

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0

Betting on a Quinella. Winning a quinella bet requires selecting thehorses that finish 1^(st) and 2^(nd) without regard to order. To bet the¾ quinella is equivalent to selecting the following outcomes

-   -   (1) Horse 3 wins, horse 4 finishes 2^(nd), and any of the        remaining horses finish 3^(rd);    -   (2) Horse 4 wins, horse 3 finishes 2^(nd), and any of the        remaining horses finish 3^(rd).        (Equivalently, the 3/4 quinella bet is a combined bet on the 3/4        exacta and the 4/3 exacta). These fundamental outcomes for        condition (1) are 17 and 18, and for condition (2) are 23        and 24. In this case, a_(j) is as follows

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1

Betting on a Trifecta. To win a trifecta bet requires selecting thehorses that finish 1^(st), 2^(nd), and 3^(rd) in order. For instancebetting the 4/3/2 trifecta is bet that horse 4 wins, horse 3 finishes2^(nd), and horse 2 finishes 3^(rd), which is fundamental outcome 24above. In this case, a_(j) is as follows

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Betting on a Boxed Trifecta. To win a boxed trifecta bet requiresselecting the horses that finish 1^(st), 2^(nd), and 3^(rd) withoutregard to order. For instance betting the 4/3/2 boxed trifecta is a betthat one of the following outcomes occurs

-   -   (1) Horse 2 wins, horse 3 finishes 2^(nd), and horse 4 finishes        3^(rd);    -   (2) Horse 2 wins, horse 4 finishes 2^(nd), and horse 3 finishes        3^(rd);    -   (3) Horse 3 wins, horse 2 finishes 2^(nd), and horse 4 finishes        3^(rd);    -   (4) Horse 3 wins, horse 4 finishes 2^(nd), and horse 2 finishes        3^(rd);    -   (5) Horse 4 wins, horse 2 finishes 2^(nd), and horse 3 finishes        3^(rd);    -   (6) Horse 4 wins, horse 3 finishes 2^(nd), and horse 2 finishes        3^(rd).        which correspond to fundamental outcomes 10, 12, 16, 18, 22 and        24 respectively. Thus,

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1

-   7. Wheeling a Horse. Wheeling a horse is a betting technique where    the bettor combines a specific horse with all other horses in a bet    such as a quinella, exacta, trifecta, or daily double (see below).    Wheeling the 3 horse in an exacta bet is a bet on the following    outcomes    -   (1) Horse 3 wins and any other horse finishes 2^(nd);    -   (2) Horse 3 finishes 2^(nd) and any other horse finishes 1^(st).        The first condition is met by fundamental outcomes 13 through 18        and the second condition is met by fundamental outcomes 3, 4, 9,        10, 23, 24.

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 0 0 1 1 0 0 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1

Betting Against a Horse. In an example embodiment, bettors can betagainst a specific horse. Betting against the 3 horse to place meansbetting that the 3 horse neither wins nor finishes 2^(nd), whichcorresponds to fundamental outcomes 1, 2, 5, 6, 7, 8, 11, 12, 19, 20,21, and 22. Thus the a vector is as follows

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) 1 1 0 0 1 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0

-   8. Different Relative Payouts. In an example embodiment a bettor can    specify a bet for instance that pays out different amounts depending    on what outcome occurs. For instance, a bet on horse 2 that pays out    twice as much money if horse 2 wins than if horse 2 finishes second    has the following a_(j) vector

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24a_(j,s) .5 .5 0 0 0 0 1 1 1 1 1 1 0 0 .5 .5 0 0 0 0 .5 .5 0 0The section below discusses how the current set of 24 fundamentaloutcomes can be expanded to accommodate other types of bets.

Betting on the Superfecta. To bet the superfecta requires picking, inorder of finish, the winner, the 2^(nd) place finisher, the 3^(rd) placefinisher, and the 4^(th) place finisher. In a four horse race, bettingthe superfecta is equivalent to a specific trifecta bet, e.g. the4/3/2/1 superfecta is equivalent to betting the 4/3/2 trifecta. Why? Ifthe 4/3/2 trifecta wins, then the 1 horse must finish 4^(th) (assumingthat all horses finish) and so the 4/3/2/1 superfecta wins. For raceswith more than 4 horses, a different set of outcomes must be created forsuperfecta wagering. For instance, for a five horse race, the wageringassociation will have to set up fundamental outcomes for the 1^(st),2^(nd), 3^(rd), and 4^(th) place finishers. In this case, followingequation 15.4.1A, h equals 5, d equals 4, and S equals 120 (120 equals5×4×3×2) such fundamental outcomes.

Betting on the Daily Double. Winning the daily double requires thebettor to pick the winner of two specific consecutive races. There areat least two ways for a wagering association to include enhancedparimutuel wagering on the daily double. First, daily double bets can beput in their own pool, as is currently done in horse wagering. Second,the set of outcomes can be combined to include two races jointly, whichwill create a large outcome space. For instance, if there are h₁ horsesfor the 1^(st) race, h₂ horses in the 2^(nd) race, and the wageringassociation allows betting on the 1^(st) three finishing horses, thenthe outcome space will be of size

$\begin{matrix}{S = {\frac{{h_{1}!}{h_{2}!}}{{\left( {h_{1} - 3} \right)!}{\left( {h_{2} - 3} \right)!}} = {{h_{1}\left( {h_{1} - 1} \right)}\left( {h_{1} - 2} \right){h_{2}\left( {h_{2} - 1} \right)}\left( {h_{2} - 2} \right)}}} & {15.4{.1}B}\end{matrix}$

A similar approach can be used for the Pick-Six, where the bettor has topick the winner of six pre-specified races.

Multiple Entry Horse Races. In certain horse races, multiple horses areentered under the same number. An example embodiment can be used forwagering in this case. In the simplest case, if there are two horsesrunning the race with the number 1, then the outcome space will beincreased to accommodate events such as the 1 horse winning and the 1horse finishing 2^(nd). For a race with two horses running with thenumber 1, one horse running with the number 2, one horse running withthe number 3, and one horse running with the number 4, then the outcomespace will include the previous 24 fundamental outcomes but also havethe following new nine fundamental outcomes listed in Table 15.4.1B.

TABLE 15.4.1B Additional fundamental outcomes in a four horse race withtwo horses with the number 1. 2^(nd) Place 3^(rd) Place FundamentalWinner of Finisher of Finisher of Outcome Horse Race Horse Race HorseRace 25 1 1 2 26 1 1 3 27 1 1 4 28 1 2 1 29 1 3 1 30 1 4 1 31 2 1 1 32 31 1 33 4 1 1More generally, the size of the outcome space with two horses runningwith the same number and betting on the first three finishers of thehorse race is

S=(h−1)(h−2)(h−3)+3(h−2)  15.4.1C

where h denotes the total number of horses in the race (h is one greaterthan the number of unique numbers for horses in the race).

15.4.2 Using Enhanced Parimutuel Wagering in Gaming Against the House

This section shows how to apply enhanced parimutuel wagering to gamingthat is normally done against the house.

15.4.2.1 Games Between Two Teams

Consider enhanced parimutuel wagering for games between two teams or twopersons, which covers a large portion of sporting events in the U.S.including baseball games, basketball games, football games, hockeygames, soccer games, boxing matches, and tennis matches.

For concreteness, consider a basketball game played between a New Yorkteam and a San Antonio team. Assume the wagering association allows forbets on which team wins and by how many points. In this case, thefundamental outcomes might be as listed in Table 15.4.2.1A.

TABLE 15.4.2.1A Fundamental outcomes for a New York versus San Antoniobasketball game. Outcome # s Fundamental Outcome 1 New York wins by 7 ormore points 2 New York wins by exactly 6 points 3 New York wins byexactly 5 points 4 New York wins by exactly 4 points 5 New York wins byexactly 3 points 6 New York wins by exactly 2 points 7 New York wins byexactly 1 points 8 New York loses by exactly 1 points 9 New York losesby exactly 2 points 10 New York loses by exactly 3 points 11 New Yorkloses by exactly 4 points 12 New York loses by exactly 5 points 13 NewYork loses by exactly 6 points 14 New York loses by 7 or more pointsOutcomes 1 and 14 are a victory or loss by the New York team of 7 ormore points, where the number 7 has been selected somewhat arbitrarily.For instance, a wagering association might wish to allow for 12 outcomesand have outcomes of a victory or loss by the New York team of 6 or morepoints.

Betting a Specific Point Spread. To bet that the New York team will winby exactly 6 points would be a bet on fundamental outcome 2. In thiscase, a_(j) is as follows

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 a_(j,s) 0 1 0 0 0 0 0 0 0 0 00 0 0

Betting a Specific Point Spread or Higher. To bet that the San Antonioteam will win by 5 or more points is a bet on fundamental outcomes 12,13, and 14, since a New York team loss by a certain number of points isa San Antonio team victory by that same number of points.

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 a_(j,s) 0 0 0 0 0 0 0 0 0 0 01 1 1

Betting on a Team to Win. To bet on the New York team to win, note thatin basketball the New York team wins if and only if they outscore theSan Antonio team by one or more points (there are no ties inbasketball). Thus, this event is thus covered by outcomes 1 through 7.

Outcome s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 a_(j,s) 1 1 1 1 1 1 1 0 0 0 00 0 0

This approach can be used for wagering on other sporting events with thefollowing modest modifications:

-   -   A baseball game where number of runs scored replaces points;    -   Football games, hockey games, and soccer games, where a        fundamental outcome is included to allow for a game to end in a        tie;    -   Any series of games between two teams (such as the World Series        in baseball), where the number of games won replaces the points;    -   A tennis match where number of sets won replaces the points;    -   A boxing match where the number of rounds fought and the overall        winner replaces the points;    -   A basketball or football game where the sum of the points scored        replaces the point differential.

15.4.2.2 Tournament Style Competition

A wagering association can use enhanced parimutuel wagering for wageringon single elimination tournaments. In single elimination tournamentsthere are scheduled rounds with the winner of each round moving on tothe next round and the loser of each round being eliminated. Thetournament ends when only one participant remains. Examples of suchtournaments include

-   -   The post season play in most major U.S. professional and        collegiate sports, including football, baseball, basketball,        hockey, and soccer;    -   Tournaments such as the U.S. Tennis open.        In single elimination tournaments, participating entities can be        teams (as in baseball where two teams play one another) or        individuals (as in singles tennis where two players play against        each other). Participants are eliminated after losing a round,        where a round can be a single game (as in football) or a “single        series” of games (as in baseball postseason where teams advance        after winning a five game or a seven game series).

In single elimination tournaments, typically the number of participantsis a power of 2. If the number of participants is 2^(h), then the numberof rounds in the tournament is h. If the wagering organization allowsbetting solely on the winner of the tournament, then the number offundamental outcomes is equal to the number of participants, which inthis case is 2^(h). However, to allow for wagering on the winner of thetournament or the results of any particular round, the number ofoutcomes S is

S=2² ^(h−1) ×2² ^(h−2) × . . . ×2²×2¹=2² ^(h) ⁻¹  15.4.2.2A

For example, in the Major League post season, two teams from theAmerican League (“AL”) play each other in the American LeagueChampionship Series (“ALCS”) and two teams from the National League(“NL”) play each other in the National League Championship Series(“NLCS”).

The winner of the ALCS and the winner of the NLCS play each other in theWorld Series. For simplicity, assume that AL East team plays against theAL West team in the ALCS, and assume that the NL East team plays againstNL West team in the NLCS.

Since there are four teams, the number of rounds in the tournament istwo and the number of fundamental outcomes S equals eight by equation15.4.2.2A. These fundamental outcomes are listed in Table 15.4.2.2A.

TABLE 15.4.2.2A The fundamental outcomes for the Major League BaseballLeague Championships and the World Series. Fundamental World SeriesOutcome ALCS Winner NLCS Winner Winner 1 AL East NL East AL East 2 ALEast NL East NL East 3 AL East NL West AL East 4 AL East NL West NL West5 AL West NL East AL West 6 AL West NL East NL East 7 AL West NL West ALWest 8 AL West NL West NL WestDifferent bets can be mapped to these fundamental outcomes.

Betting on a Team to Win a Championship Series. To bet on the AL Westteam to win the ALCS is a bet on fundamental outcomes 5, 6, 7 and 8.Thus a_(j)=[0 0 0 0 1 1 1 1].

Betting on a Team Winning the World Series. To bet on the NL East teamto win the World Series is a bet on fundamental outcomes 2 and 6 soa_(j)=[0 1 0 0 0 1 0 0].

Betting on a Team Losing in the World Series. To bet on the NL West teamto win the NLCS and lose in the World Series is a bet on fundamentaloutcomes 3 and 7 so a_(j)=[0 0 1 0 0 0 1 0].

Betting on a Team to NOT Win the World Series. To bet on the AL Eastteam not to win the World Series is a bet on fundamental outcomes 2, and4 through 8 so a_(j)=[0 1 0 1 1 1 1 1].

When applying enhanced parimutuel wagering to tournaments with a largernumber of teams, the outcome space grows very large very quickly. Forinstance, the outcome space for the NCAA basketball tournament with 64teams is (using equation 15.4.2.2A with h=6=log₂64)

S=2² ^(h) ⁻¹=2⁶³≈9.2×10¹⁸  15.4.2.2B

To lower the size of the outcome space for this tournament, the wageringassociation may wish to create a small number of separate pools withsmaller outcome spaces. One such pool might be a pool to wager on teamsto win at least 4 games (i.e. “make it to the final four”), which has anoutcome space of 16⁴ or 65,536 outcomes.

15.4.2.3 Other Multi-Participant Competitions

In addition to single elimination tournaments, a wagering associationmay set up wagering on other events and competitions with more than twoparticipants including the following

-   -   The winner of the American League East in Major League Baseball        in 2003;    -   The NFL player with the most rushing yards in 2003;    -   The golfer with the highest earnings in 2003;    -   The winner of a NASCAR race, a golf tournament, or the Tour de        France.        If the wagering association allows only for wagering on the        winner of the event, then the size of the outcome space will be        the number of possible winners and each outcome will correspond        to a participant winning. For instance, there are five teams in        the American League East in baseball. Thus wagering on the        winner of the American League East has an outcome space of size        S equals five.

15.4.2.4 Roulette

A wagering association can apply enhanced parimutuel wagering in casinogames such as roulette. For roulette, the size of the outcome space is Sequals 38. Based on such an outcome space, a bettor can make bets onspecific number (e.g. 1, 2, 3, . . . , 36), a color (red, black), or aspecific set of numbers (e.g. even versus odd).

15.4.3 Using Enhanced Parimutuel Wagering in Lotteries

Wagering associations can employ enhanced parimutuel wagering forlotteries, giving bettors control over whether they buy specific lottotickets based on the payout of the ticket.

For example, consider a Lottery Daily Numbers game, which pays out basedon an integer drawn at random between 0 and 999. In this case, the sizeof the outcome space is S equals 1,000 and each outcome corresponds to apossible number.

A Straight Play. For a straight play, the bettor selects a three-digitnumber and wins if that outcome occurs. In this case, a_(j) equals 0 in999 locations and equals 1 in one location.

A Box Play. For a box play, the bettor selects a three-digit number inwhich two digits are the same. If the bettor selects the number 122,then the bettor wins if the numbers 122, 212, or 221 are drawn. In thiscase, a_(i) equals 0 in 997 locations and equals 1 in three locations.

Appendix: Notation Used in Section 15

a_(j): a vector representing the weight for the fundamental bets for betj, j=1, 2, . . . , J;a_(j,s): a scalar representing the weight for fundamental bet s for betj, s=1, 2, . . . , S and j=1, 2, . . . , J;j: a scalar used to index the bets j=1, 2, . . . , J;J: a scalar representing the total number of bets;M: a scalar representing the total cleared premium;o_(f): a scalar representing the limit odds to 1 for bet j, j=1, 2, . .. , J;p_(s): a scalar representing the final price of the sth fundamental bets=1, 2, . . . , 5;r_(j): a scalar representing the requested maximum payout for bet j,j=1, 2, . . . , J, where bet j is a payout bet;s: a scalar used to index across fundamental outcomes or fundamentalbets;S: a scalar representing the number of fundamental outcomes orfundamental bets;u_(j)a scalar representing the requested premium amount for bet j, j=1,2, . . . , J, where bet j is a premium bet;v_(j): a scalar representing the final filled premium amount for bet j,j=1, 2, . . . , J;w_(j): a scalar representing the limit price for bet j, j=1, 2, . . . ,J;x_(j): a scalar representing the final filled payout amount for bet j,j=1, 2, . . . , J;y_(s): a scalar representing the aggregate filled payouts forfundamental bet s for s=1, 2, . . . , S;θ_(s): a scalar representing the invested premium amount for fundamentalbet s, s=1, 2, . . . , S;ω_(d): a scalar representing the final odds to 1 for the outcomesassociated with bet j, j=1, 2, . . . , J;π_(j): a scalar representing the final price of bet j, j=1, 2, . . . ,J.

16. TECHNICAL APPENDIX

This technical appendix provides the mathematical foundation underlyingthe computer, code listing of Table 1: Illustrative Visual BasicComputer Code for Solving CDRF 2. That computer code listing implementsa procedure for solving the Canonical Demand Reallocation Function (CDRF2) by preferred means which one of ordinary skill in the art willrecognize are based upon the application of a mathematical method knownas fixed point iteration.

As previously indicated in the specification, the simultaneous systemembodied by CDRF 2 does not provide an explicit solution and typicallywould require the use of numerical methods to solve the simultaneousquadratic equations included in the system. In general, such systemswould typically be solved by what are commonly known as “grid search”routines such as the Newton-Raphson method, in which an initial solutionor guess at a solution is improved by extracting information from thenumerical derivatives of the functions embodied in the simultaneoussystem.

One of the important advantages of the demand-based trading methods ofthe present invention is the careful construction of CDRF 2 which allowsfor the application of fixed point iteration as a means for providing anumerical solution of CDRF 2. Fixed point iteration means are generallymore reliable and computationally less burdensome than grid searchroutines, as the computer code listing in Table 16.1 illustrates.

Fixed Point Iteration

The solution to CDRF 2 requires finding a fixed point to a system ofequations. Fixed points represent solutions since they convey theconcept of a system at “rest” or equilibrium, i.e., a fixed point of asystem of functions or transformations denoted g(a) exists if

a=g(a)

Mathematically, the function g(a) can be said to be a map on the realline over the domain of a. The map, g(x), generates a new point, say, y,on the real line. If x=y, then x is called a fixed point of the functiong(a). In terms of numerical solution techniques, if g(a) is a non-linearsystem of equations and if x is a fixed point of g(a), then a is alsothe zero of the function. If no fixed points such as x exist for thefunction g(a), then grid search type routines can be used to solve thesystem (e.g., the Newton-Raphson Method, the Secant Method, etc.). If afixed point exists, however, its existence can be exploited in solvingfor the zero of a simultaneous non-linear system, as follows.

Choose an initial starting point, x₀, which is believed to be somewherein the neighborhood of the fixed point of the function g(a). Then,assuming there does exist a fixed point of the function g(a), employ thefollowing simple iterative scheme:

x _(i−1) =g(x _(i)), where x₀ is chosen as starting point

where i=0, 1, 2, . . . n. The iteration can be continued until a desiredprecision level,ε, is achieved, i.e.,

x _(n) =g(x _(n−1)), until |g(x _(n−1))−x _(n)|<ε

The question whether fixed point iteration will converge, of course,depends crucially on the value of the first derivative of the functiong(x) in the neighborhood of the fixed point as shown in FIG. 69.

As previously indicated, an advantage of the present invention is theconstruction of CDRF 2 in such a way so that it may be represented interms of a multivariate function, g(x), which is continuous and has aderivative whose value is between 0 and 1, as shown below.

Fixed Point Iteration as Applied to CDRF 2

This section will demonstrate that (1) the system of equations embodiedin CDRF 2 possesses a fixed point solution and (2) that this fixed pointsolution can be located using the method of fixed point iterationdescribed in Section A, above. The well known fixed point theoremprovides that, if g:[a, b]−>[a, b] is continuous on [a, b] anddifferentiable on (a, b) and there is a constant k<1 such that for all xin (a, b),

|g′(x)|≦k

then g has a unique fixed point x* in [a, b]. Moreover, for any x in [a,b] the sequence defined by

x ₀ =x and x _(n+1) =g(x _(n))

converges to x* and for all n

${{x_{n} - x^{*}}} \leq {\frac{k^{n}*{{x_{1} - x_{0}}}}{1 - k}.}$

The theorem can be applied CDRF 2 as follows. First, CDRF 2 in apreferred embodiment relates the amount or amounts to be invested acrossthe distribution of states for the CDRF, given a payout distribution, byinverting the expression for the CDRF and solving for the traded amountmatrix A:

A=P*Π(A,ƒ)⁻¹  (CDRF 2).

CDRF 2 may be rewritten, therefore, in the following form:

A=g(A)

where g is a continuous and differentiable function. By theaforementioned fixed point theorem, CDRF 2 may be solved by means offixed point iteration if:

g′(A)<1

i.e., the multivariate function g(A) has a first derivative less than 1.Whether g(A) has a derivative less than 1 with respect to A can beanalyzed as follows. As previously indicated in the specification, forany given trader and any given state i, CDRF2 contains equations of thefollowing form relating the desired payout p (assumed to be greater than0) to the traded amount a required to generate the desired payout, givena total traded amount already traded for state i of T_(i) (also assumedto be greater than 0) and the total traded amount for all the states ofT:

$\alpha = {\left( \frac{T_{i} + \alpha}{T + \alpha} \right)*p}$

so that

${g(\alpha)} = {\left( \frac{T_{i} + \alpha}{T + \alpha} \right)*p}$

Differentiating g(a) with respect to α yields:

${g^{\prime}(\alpha)} = {\left( \frac{T - T_{i}}{T + \alpha} \right)*\frac{p}{T + \alpha}}$

Since the DRF Constraint defined previously in the specificationrequires that payout amount p not exceed the total amount traded for allof the states, the following condition holds:

$\frac{p}{T + \alpha} \leq 1$

and therefore since

$\left( \frac{T - T_{i}}{T + \alpha} \right) < 1$

it is the case that

0<g′(α)<1

so that the solution to CDRF 2 can be obtained by means of fixed pointiteration as embodied in the computer code listing of Table 1.

17. CONCLUSION

Example embodiments of the invention have been described in detailabove, various changes thereto and equivalents thereof will be readilyapparent to one of ordinary skill in the art and are encompassed withinthe scope of this invention and the appended claims. For example, manytypes of demand reallocation functions (DRFs) can be employed to financegains to successful investments with losses from unsuccessfulinvestments, thereby achieving different risk and return profiles totraders. Additionally, this disclosure has discussed methods and systemsfor replicated derivatives strategies and financial products, as well asfor groups and portfolios of DBAR contingent claims, and markets andexchanges and auctions for those strategies, products and groups. Themethods and systems of the present invention can readily be adapted byfinancial intermediaries for use within the traditional capital andinsurance markets. For example, a group of DBAR contingent claims can beembedded within a traditional security, such as a bond for a givencorporate issuer, and underwritten and issued by an underwriter aspreviously discussed. It is also intended that such embodiments andtheir equivalents are encompassed by the present invention and theappended claims.

The present invention has been described above in the context of tradingderivative securities, specifically the implementation of an electronicderivatives exchange which facilitates the efficient trading of (i)financial-related contingent claims such as stocks, bonds, andderivatives thereon, (ii) non-financial related contingent claims suchas energy, commodity, and weather derivatives, and (iii) traditionalinsurance and reinsurance contracts such as market loss warranties forproperty-casualty catastrophe risk. The present invention has also beendescribed above in the context of a DBAR digital options exchange, andin the context of offering DBAR-enabled financial products andderivatives strategies. The present invention has also been describedabove in the context of an enhanced parimutuel wagering system on abetting pool on an underlying event (for example, a horse or dog race, asporting event or the lottery), and can be applied to running one ormore betting pools on one or more underlying events. The presentinvention is not limited to these contexts, however, and can be readilyadapted to any contingent claim relating to events which are currentlyuninsurable or unhedgable, such as corporate earnings announcements,future semiconductor demand, and changes in technology.

In the preceding specification, the present invention has been describedwith reference to specific exemplary embodiments thereof. It will,however, be evident that various modifications and changes may be madethereunto without departing from the broader spirit and scope of thepresent invention as set forth in the claims that follow. Thespecification and drawings are accordingly to be regarded in anillustrative rather than restrictive sense.

1. A computer-implemented wagering method, comprising: for each of atleast two placed bets of different type on at least one race,allocating, by a processor, portions of a premium specified for therespective placed bet to a respective subset of a plurality ofpredefined fundamental bets determined to be equivalent to therespective placed bet, the plurality of fundamental bets correspondingto a plurality of predefined mutually exclusive fundamental outcomes;and executing the respective subsets of bets.
 2. The method according toclaim 1, wherein odds of the each of the at least two placed bets isaffected by the other of the each of the at least two placed bets. 3.The method according to claim 1, wherein the at least one race is ahorse race.
 4. The method according to claim 3, where one of the atleast two placed bets is one of (a) against a horse, (b) on a horse tofinish second without specifying any horse to finish first, (c) on ahorse to finish third without specifying any horse to finish first orsecond, and (d) on a horse to finish second or third.
 5. The methodaccording to claim 1, further comprising: recording respective firstfundamental bets on all of the plurality of fundamental outcomes by asingle party.
 6. The method according to claim 5, wherein, subsequent tothe recordation of the first fundamental bets, premiums are applied forsubsequent bets to the fundamental outcomes such that, for eachfundamental outcome, should the respective outcome occur, a total payoutfor the subsequent bets on the respective outcome is equal to a sum ofall premiums applied to all of the fundamental bets divided by aquotient of a total applied premium for all of the first fundamentalbets divided by a total of applied premium to all of the subsequentfundamental bets.
 7. The method according to claim 1, wherein:throughout a betting period, a cost for a single unit of each of theplurality of fundamental bets is set such that a sum of the costs equalsone predefined unit of a predefined currency; and each of the singleunits of each of the plurality of fundamental bets produces a payout ofthe one predefined unit of the predefined currency if the respectivefundamental outcome of the respective fundamental bet occurs.
 8. Themethod according to claim 1, wherein, odds are specifiable for theplaced bets, and, for each of the placed bets for which odds arespecified, none of a specified premium is applied conditional upon thatodds on the placed bet are less than the specified odds, all of thespecified premium is applied conditional upon that the odds on theplaced bet are better than the specified odds, and some and not all ofthe specified premium is applied conditional upon that the odds on theplaced bet equal the specified odds.
 9. The method according to claim 8,wherein, for each of at least one of the placed bets, an extent to whicha specified premium for the placed bet is applied varies throughout thebetting period.
 10. The method according to claim 9, wherein: uponreceipt of a newly placed bet, an iterative process is performed fordetermining an extent to which to apply a specified premium of the newlyplaced bet; and in each iteration: odds for the newly placed bet aredetermined based on an extent to which specified premiums of priorplaced bets have been applied; and an extent to which specified premiumsof one or more of the prior placed bets are applied are redeterminedbased on a modification of odds for the one or more prior placed betsdue to the extent to which the specified premium for the newly placedbet is applied, the iterative process continuing until an equilibrium,within a predefined tolerance, is reached.
 11. The method according toclaim 1, wherein the two placed bets are on two different racesoccurring at different times.
 12. A computer-implemented wageringmethod, comprising: prior to an end of a betting period in which betsare recordable on a horse racing event, calculating and outputting, by acomputer processor, indicative odds of a bet that a horse in the racingevent will finish first or second in the racing event, the bet beingsuch that a payout on the bet is receivable based on results of theevent regardless of whether the horse finishes first or second in theevent.
 13. The method according to claim 12, wherein the racing event isa horse racing event.
 14. A computer-implemented wagering method,comprising: for a placed bet on a complex outcome of a racing event, acomputer processor allocating portions of a premium specified for theplaced bet to a subset of a plurality of predefined fundamental bets onpredefined fundamental and mutually exclusive outcomes determined by theprocessor to be equivalent to the placed bet, wherein odds vary betweenthe subset of fundamental bets, such that a payout on the placed betdiffers depending on which of the mutually exclusive outcomes occur. 15.The method according to claim 14, wherein the racing event is a horseracing event.
 16. A computer-implemented wagering method, comprising:receiving, by a processor, a request for placement of a bet on a racingevent, the request specifying a premium and odds; determining, by theprocessor, an extent of the specified premium to execute according topredefined conditions that none of the specified premium is appliedconditional upon that odds on the requested bet are less than thespecified odds, all of the specified premium is applied conditional uponthat the odds on the requested bet are better than the specified odds,and some and not all of the specified premium is applied conditionalupon that the odds on the requested bet equal the specified odds; andhandling placement of the bet, by the processor, in accordance with thedetermination.